Major Terms and Results - Math 301 Real Analysis - Fall 2017

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Be sure to check back, because there will certainly be some changes during the semester.

All numbers indicate sections from Understanding Analysis, Second Edition by Abbott.

Chapter 1 The Real Numbers

Theorem 1.1.1. There is no rational number whose square is 2.

Theorem 1.2.6. Two real numbers \( a\) and \( b\) are equal iff for every real number \( \e>0\) it follows that \( |a-b| <\e \).

Principle of Induction. If \(S\subset \N\) such that

1. \( S\) contains 1 and
2. whenever \( S\) contains \( n\in\N \), it also contains \( n+1\),

Axiom of Completeness. Every nonempty set of real numbers that is bounded above has a least upper bound.

Definition 1.3.1. A set \( A\subset \R\) is bounded above if there exists a number \( b\in\R\) such that \( a\le b\) for all \( a\in A\). The number \(b\) is called an upper bound for \(A\).

Similarly, the set \(A\) is bounded below if there exists a lower bound \( l\in\R\) satisfying \( l\le a\) for every \( a\in A\).

Definition 1.3.2. A real number \(s\) is the least upper bound, or supremum for a set \( A\subset \R\) if it meets the following two criteria:

1. \( s\) is an upper bound for \( A\)
2. if \( b\) is any upper bound for \( A\), then \( s \le b \)

Lemma 1.3.8. Suppose \(s\in\R\) is an upper bound for \(A\subset\R\). Then \(s=\sup A\) iff \(\forall \e>0\ \exists\ a\in A\ \ni s-\e < a\).

Theorem 1.4.1 (Nested Interval Property). For each \(n\in \N\), assume that we have a closed interval \(I_n=[a_n,b_n]\) such that \[ I_1 \supset I_2 \supset I_3 \supset I_4 \supset \cdots\] Then \(\displaystyle \bigcap_{n=1}^{\infty} I_n \ne \emptyset \)

Theorem 1.4.2 (Archimedean Property).

1. Given any \(x\in\R\), there exists \(n\in\N\) such that \(n>x\).
2. Given any real number \(y>0\), there exists \(n\in\N\) such that \(\frac{1}{n} < y\).

Theorem 1.4.3 (Density of \( \Q\) in \(\R\)). For every two real numbers \(a\) and \(b\) where \(a< b\), there exists a rational \(r\) satisfying \(a< r < b\).

Corollary 1.4.4. Given any two real numbers \( a < b\), there exists an irrational number \( t\) satifying \( a < t < b\).

Theorem 1.4.5 . There exists a real number \( \alpha\in\R\) satisfying \( \alpha^2 = 2\).

Definition 1.5.1. A function \( f:A\to B\) is one-to-one if \( a_1 \ne a_2\) in \( A\) implies that \( f(a_1) \ne f(a_2)\) in \(B\). The function is onto if, given any \(b\in B\), it is possible to find an element \( a\in A\) for which \(f(a)=b\).

Definition 1.5.2. A set \(A\) has the same cardinality as \(B\) if there exists \(f:A\to B\) that is one-one and onto. In this case we write \( A\sim B\).

Definition 1.5.5. A set \(A\) is countable if \( \N \sim A\). An infinite set that is not countable is called an uncountable set.

Theorem 1.5.6.

1. The set \(\Q\) is countable.
2. The set \(\R\) is uncountable.

Theorem 1.5.7. If \(A\subset B\) and \(B\) is countable, then \(A\) is either countable or finite.

Theorem 1.5.8.

1. If \(A_1,A_2, \ldots,A_m\) are each countable sets, then the union \(A_1\cup A_2\cup\cdots\cup A_m\) is countable.
2. If \(A_n\) is a countable set for each \(n\in\N\), then \(\dst\cup_{n=1}^\infty A_n\) is countable.

Theorem 1.6.1. The open interval \( (0,1)\subset\R \) is uncountable.

Theorem 1.6.2 (Cantor's Theorem). Given any set \( A\), there does not exist a function \( f:A\to P(A)\) that is onto.

Chapter 2 Sequences and Series

Definition 2.2.3. A sequence \( (a_n) \) converges to a real number \( a\) if for every \( \e > 0 \) there exists \( N\in\N \) such that \( n\ge N\) implies \( | a_n - a | < \e\).

Definition 2.2.4. Given a real number \( a\in \R\) and a positive number \( \e>0\), the set \[ V_{\e}(a) = \{ x\in\R \mid |x-a| < \e \} \] is called the \( \e\)-neighborhood of \( a\) .

Definition 2.2.3B (Convergence of a Sequence: Topological Version). A sequence \( (a_n)\) converges to a if, given any \(\epsilon\)-neighborhood \( V_{\e}(a)\) of a, there exists a point in the sequence after which all of the terms are in \( V_{\e}(a)\). In other words, every \(\epsilon\)-neighborhood contains all but a finite number of the terms of \( (a_n)\).

Theorem 2.3.2. Every convergent sequence is bounded.

Theorem 2.3.3 (Algebraic Limit Theorem). Let \( \lim a_n=a \) and \( \lim b_n=b \). Then

1. \( \lim (ca_n) = ca\) for all c\( \in\R\)

2. \( \lim(a_n+b_n) = a+b \)

3. \( \lim(a_nb_n)=ab\)

4. \( \lim\left(\frac{a_n}{b_n}\right)=\frac{a}{b}\) if \( b\ne 0\)

Theorem 2.3.4 (Order Limit Theorem). Let \( \lim a_n=a \) and \( \lim b_n=b \). Then

1. If \( a_n\ge 0\) for all \( n\in\N\), then \( a\ge 0\)

2. If \( a_n \le b_n\) for all \( n\in\N\), then \( a\le b\)

3. If there exists \( c\in\R\) such that \(c\le b_n\) for all \(n\in\N\), then \(c\le b\)
   Similarly, if \( a_n\le c\) for all \( n\in\N\), then \( a\le c\)

Theorem 2.4.2 (Monotone Convergence Theorem). If a sequence is monotone and bounded, then it converges.

Definition 2.4.3 (Convergence of a Series). Let \( (b_n) \) be a sequence. An infinite series is a formal expression of the form \[ \sum_{n=1}^\infty b_n = b_1 + b_2 + b_3 + b_4 + \cdots \] We define the corresponding sequence of partial sums \( (s_m)\) by \[ s_m = b_1 + \cdots + b_m \] and say that the series \( \sum_{n=1}^\infty b_n\) converges to \( B\) if the sequence \( (s_m)\) converges to \( B\). In this case we write \( \sum_{n=1}^\infty b_n = B\).

Theorem 2.4.6 (Cauchy Condensation Test). Suppose that \( (b_n) \) is decreasing and satisfies \( b_n\ge 0\) for all \( n\). Then \( \sum_{n=1}^\infty b_n\) converges iff the series \[ \sum_{n=0}^\infty 2^n\ b_{2^n} = b_1 + 2b_2 + 4b_4 + 8b_8 + 16b_{16} + \cdots \] converges.

Corollary 2.4.7. The series \(\dst \sum_{n=1}^\infty \frac{1}{n^p}\) converges iff \( p>1\).

Definition 2.5.1. Let \( (a_n) \) be a sequence of real numbers, and let \( n_1 < n_2 < n_3<\cdots\) be an increasing sequence of natural numbers. Then the sequence \[ (a_{n_1}, a_{n_2}, a_{n_3}, \ldots ) \] is called a subsequence of \( a_n\) and is denoted by \( (a_{n_k})\) where \( k\in\N\) indexes the subsequence.

Theorem 2.5.2. Subsequences of a convergent sequence converge to the same limit as the original sequence.

Theorem 2.5.5 (Bolzano-Weierstrass Theorem). Every bounded sequence contains a convergent subsequence.

Definition 2.6.1. A sequence \( (a_n)\) is a Cauchy sequence iff for every \( \e>0\) there exists \( N\in\N\) such that \( m\ge N\) and \( n\ge N\) imply \(|a_n-a_m|<\e\).

Theorem 2.6.2. Every convergent sequence is a Cauchy sequence.

Lemma 2.6.3. Cauchy sequences are bounded.

Theorem 2.6.4 (Cauchy Criterion). A sequence converges if and only if it is a Cauchy sequence.

Theorem 2.7.1 (Algebraic Limit Theorem for Series). If \(\sum_{k=1}^\infty a_k = A\) and \(\sum_{k=1}^\infty B_k = B\), then

1. \(\sum_{k=1}^\infty c a_k = cA\) for all \( c\in\R\)
   
   
2. \(\sum_{k=1}^\infty a_k + b_k = A+B\)

Theorem 2.7.2 (Cauchy Criterion for Series). The series \(\sum_{k=1}^\infty a_k\) converges iff given \( \e>0\), there exists \(N\in\N\) such that whenever \(n>m\ge N\) it follows that \[ |a_{m+1} + a_{m+2}+\cdots+a_n|<\e \]

Theorem 2.7.3. If the series \(\sum_{k=1}^\infty a_k \) converges, then \( (a_k)\to 0\).

Theorem 2.7.4 (Comparison Test). Assume \( (a_k)\) and \( (b_k) \) are sequences satisfying \( 0\le a_k \le b_k\) for all \( k\in\N\).

1. If \(\sum_{k=1}^\infty b_k \) converges, then \(\sum_{k=1}^\infty a_k \) converges.
   
   
2. If \(\sum_{k=1}^\infty a_k \) diverges, then \(\sum_{k=1}^\infty b_k \) diverges.

Theorem 2.7.6(Absolute Convergence Test). If the series \(\sum_{k=1}^\infty |a_k| \) converges, then \(\sum_{k=1}^\infty a_k \) converges as well.

Theorem 2.7.7(Alternating Series Test). Let \( (a_n)\) be a sequence satisfying

1. \( a_1 \ge a_2 \ge a_3 \ge \cdots \) and
2. \( (a_n)\to 0 \)

Definition 2.7.8. If \(\sum_{n=1}^\infty |a_n| \) converges, then we say that \(\sum_{n=1}^\infty a_n \) converges absolutely .

If \(\sum_{n=1}^\infty a_n \) converges but \(\sum_{n=1}^\infty |a_n| \) diverges, then we say that the original series \(\sum_{n=1}^\infty a_n \) converges conditionally.

Definition 2.7.9. Let \(\sum_{k=1}^\infty a_k \) be a series. A series \(\sum_{k=1}^\infty b_k \) is called a rearrangement of \(\sum_{k=1}^\infty a_k \) if there exists a one-to-one, onto function \( f:\N\to\N\) such that \( b_{f(k)} = a_k\) for all \(n\in\N\).

Theorem 2.7.10. If a series converges absolutely, then any rearrangement of the series converges to the same limit.

Chapter 3 Basic Topology of \(\R\)

Definition 3.2.1. A set \( O\subset\R\) is open if for all points \( a\in O\) there exists an \( \e\)-neighborhood of \( V_{\e}(a)\subset O\).

Theorem 3.2.3.

1. The union of an arbitrary collectxion of open sets is open.
2. The intersection of a finite collection of open sets is open.

Definition 3.2.4. A point x is a limit point of a set A if every \(\e\)-neighborhood \(V_{\e}(x)\) intersects the set A at some point other than x.

Theorem 3.2.5. A point x is a limit point of a set A iff \( x=\lim a_n\) for some sequence \( (a_n)\) contained in A satisfying \(a_n\ne x\) for all \(n\in\N\).

Definition 3.2.6. A point \(a\in A\) is an isolated point of A if it is not a limit point of A.

Definition 3.2.7. A set \(F\subset\R\) is closed if it contains its limit points.

Theorem 3.2.8. A set \(F\subset\R\) is closed iff every Cauchy sequence contained in \(F\) has a limit that is also an element of \(F\).

Theorem 3.2.10 (Density of \(\Q\) in \(\R\)). For every \(y\in\R\), there exists a sequence of rational numbers that converges to \(y\).

Definition 3.2.11. Given a set \(A\subset\R\), let \(L\) be the set of all limit points of \(A\). The closure of \(A\) is defined to be \(\overline{A}=A\cup L\).

Theorem 3.2.12. For any \(A\subset\R\), the closure \(\overline{A}\) is a closed set and is the smallest closed set containing \(A\).

Theorem 3.2.13. A set \(O\) is open iff its complement \(O^c\) is closed. Likewise, a set \(F\) is closed iff its complement \(F^c\) is open.

Theorem 3.2.14.

1. The union of finite collection of closed sets is closed.
2. The intersection of an arbitrary collection of closed sets is closed.

Definition 3.3.1. A set \(K\subset\R\) is compact if every sequence in \(K\) has a subsequence that converges to a limit that is also in \(K\).

Theorem 3.3.4. A set \(K\subset\R\) is compact iff it is closed and bounded.

Definition 3.3.6. Let \(A\subset\R\). An open cover for \(A\) is a (possibly infinite) collection of open sets \(\{O_\lambda : \lambda \in \Lambda\}\) such that \( A\subset \bigcup_{\lambda \in \Lambda} O_\lambda\).

Given an open cover \(\{O_\lambda \}\) for \(A\), a finite subcover is a finite collection \(\{ O_1, \ldots, O_n\}\) that still covers \(A\).

Theorem 3.3.8 (Heine-Borel Theorem). Let \(K\subset\R\). The following are equivalent:

1. \(K\) is compact.

2. \(K\) is closed and bounded.

3. Every open cover for \(K\) has a finite subcover.

Chapter 4 Functional Limits and Continuity

Definition 4.2.1. Let \(f:A\to\R\), and let \(c\) be a limit point of \(A\). We say \(\dst\lim_{x\to c} f(x)=L\) if for all \(\e>0\) there exists \(\d>0\) such that \( 0<|x-c|<\d\) (and \(x\in A\)) implies \( |f(x)-L| < \e\).

Definition 4.2.1B. Let \(f:A\to\R\), and let \(c\) be a limit point of \(A\). We say \(\dst\lim_{x\to c} f(x)=L\) if for every \(\e\)-neighborhood \(V_{\e}(L)\) of \(L\), there exists a \(\d\)-neighborhood \(V_{\d}(c)\) such that for all \(x\in V_{\d}(c), x\in A, x\neq c\) implies that \( f(x)\in V_{\e}(L)\).

Theorem 4.2.3 (Sequential Criterion for Functional Limits). Let \( f:A\to \R\) and let \(c\) be a limit point of \(A\).

Corollary 4.2.5 (Divergence Criterion for Functional Limits). Let \(f:A\to\R\) and let \(c\) be a limit point of \(A\).

If there exist two sequences \( (x_n)\) and \( (y_n)\) in \(A\) where \(x_n\ne c\) and \( y_n\ne c\) and \[ \lim x_n=\lim y_n = c \text{ but } \lim f(x_n)\ne \lim f(y_n) \] then the limit \( \dst\lim_{x\to c} f(x)\) does not exist.

Definition 4.3.1 (Continuity). A function \(f:A\to\R\) is continuous at a point \(c\in A\) if for all \(\e>0\) there exists a \(\d>0\) such that \(|x-c|<\d\) (and \(x\in A\)) implies \(|f(x)-f(c)|<\e\).

If \(f\) is continuous at every point in the domain \(A\), then we say that \(f\) is continuous on \(A\).

Theorem 4.3.2. Let \(f:A\to\R\) and let \(c\in A\). The function \(f\) is continuous at \(c\) iff any one of the following three conditions is met.

1. For all \(\e>0\) there exists a \(\d>0\) such that \(|x-c|<\d\) (and \(x\in A\)) implies \(|f(x)-f(c)|<\e\);

2. For all \(V_\e(f(c))\), there exists a \(V_\d(c)\) such that \(x\in V_\d(c)\) (and \(x\in A\)) implies \(f(x)\in V_\e(f(c))\);

3. If \( (x_n)\to c\) (with \(x_n\in A\)), then \( f(x_n)\to f(c)\)

If \(c\) is a limit point of \(A\), then the above conditions are equivalent to

4. \( \lim_{x\to c} f(x) = f(c)\)

Corollary 4.3.3. Let \(f:A\to \R\) and let \(c\in A\) be a limit point of \(A\). If there exists a sequence \( (x_n)\) in \(A\) where \( (x_n)\to c\) but \( (f(x_n) ) \) does not converge to \(f(c)\), then \(f\) is not continuous a \(c\).

Theorem 4.3.4 (Algebraic Continuity Theorem). Assume \(f:A\to\R\) and \(g:A\to\R\) are continuous at a point \(c\in A\). Then

1. \( k\, f(x)\) is continuous at \(c\) for all \(k\in \R\)

2. \( f(x)+g(x)\) is continuous at \(c\)

3. \(f(x)g(x)\) is continuous at \(c\)

4. \(f(x)/g(x)\) is continuous at \(c\) provided the quotient is defined.

Theorem 4.3.9. Given \(f:A\to\R\) and \(g:B\to\R\), assume that the range of \(f\) is contained in the domain of \(g\) so that the composition \( g\circ f(x) = g(f(x))\) is defined on \(A\).

If \(f\) is continuous at \(c\in A\) and if \(g\) is continuous at \(f(c)\in B\), then \(g\circ f\) is continuous at \(c\).

Theorem 4.4.1. Let \(f:A\to\R\) be continuous and let \(K\subset A\) be compact. Then \(f(K)\) is compact.

Theorem 4.4.2 (Extreme Value Theorem). If \(f:K\to\R\) is continuous on the compact set \(K\), then \(f\) attains a maximum and minimum value on \(K\).

Definition 4.4.4. A function \(f:A\to\R\) is uniformly continuous on \(A\) iff for all \(\e>0\) there exists \(\d>0\) such that \(|x-y|<\d\) implies that \(|f(x)-f(y)| <\e\).

Theorem 4.4.5. A function \(f:A\to\R\) is not uniformly continuous on \(A\) iff there exists a particular \(\e_0>0\) and two sequences \((x_n)\) and \((y_n)\) in \(A\) such that \[ \lim |x_n-y_n| = 0 \text{ but } |f(x_n)-f(y_n)| \ge \e_0 \text{ for all } n \]

Theorem 4.4.7. If \(f:K\to\R\) is continuous and \(K\) is compact, then \(f\) is uniformly continuous on \(K\).

Theorem 4.5.1 (Intermediate Value Theorem). Suppose \(f:[a,b]\to\R\) is continuous on \([a,b]\) and \(L\in\R\) such that \[ f(a) < L < f(b) \;\;\; \text{ or } \;\;\; f(b) < L < f(a) \] Then there exists \( c\in(a,b)\) such that \( f(c)=L \).

Chapter 5 The Derivative

Definition 5.2.1. Let \(g:A\to\R\) be a function defined on an interval \(A\). Given \(c\in A\), define the derivative of \(g\) at \(c\) by \[ g'(c)=\lim_{x\to c} \frac{f(x)-f(c)}{x-c}\]

Theorem 5.2.3. If \(g:A\to\R\) is differentiable at \(c\in A\), then \(g\) is continuous at \(c\).

Theorem 5.2.4 (Algebraic Differentiability Theorem). Let \(f\) and \(g\) be functions defined on an interval \(A\), and assumbe both are differentiable at some point \(c\in A\). Then,

1. \( (f+g)'(c) = f'(c) + g'(c) \)

2. \( (kf)'(c) = k\,f'(c)\) for all \(k\in\R\)

3. \( (fg)'(c) = f'(c)g(c)+f(c)g'(c)\)

4. \( (f/g)'(c) =\dst\frac{g(c)f'(c)-f(c)g'(c)}{(g(c))^2}\) provided that \(g(c)\ne 0\)

Theorem 5.2.5 (The Chain Rule). Let \(f:A\to\R\) and \(g:B\to\R\) where \(f(A)\subset B\). If \(f\) is differentiable at \(c\in A\) and \(g\) is differentiable at \(f(c)\in B\), then \(g\circ f\) is differentiable at \(c\) with \[ (g\circ f)'(c) = g'(f(c))\ f'(c) \]

Theorem 5.2.6. Let \(f\) be differentiable on \((a,b)\). If \(f\) attains a maximum (or minimum) value at some point \(c\in(a,b)\), then \(f'(c)=0\).

Theorem 5.2.7 (Darboux's Theorem). If \(f\) is differentiable on an interval \([a,b]\), and if \(\alpha\) satisfies \[f'(a)<\alpha < f'(b) \ \ \text{ or }\ \ f'(a) > \alpha>f'(b) \] then there exists a point \(c\in(a,b)\) where \(f'(c)=\alpha\).

Theorem 5.3.1 (Rolle's Theorem). Let \(f:[a,b]\to \R\) be continuous on \([a,b]\) and differentiable on \((a,b)\). If \(f(a)=f(b)=0\), then there exists \(c\in(a,b)\) such that \(f'(c)=0\).

Theorem 5.3.2 (Mean Value Theorem). If \(f:[a,b]\to\R\) is continuous on \([a,b]\) and differentiable on \((a,b)\), then there exists \(c\in(a,b)\) such that \[ f'(c)=\frac{f(b)-f(a)}{b-a}\]

Corollary 5.3.3. If \(g:A\to\R\) is differentiable on an interval \(A\) such that \(g'(x)=0\) for all \(x\in A\), then \(g(x)=k\) for some constant \(k\in\R\).

Corollary 5.3.4. If \(f\) and \(g\) are differentiable on an interval \(A\) where \(f'(x)=g'(x)\) for all \(x\in A\), then \(f(x)=g(x)+k\) for some constant \(k\in \R\).

Theorem 5.3.5 (Generalized Mean Value Theorem). If \(f\) and \(g\) are continuous on \([a,b]\) and differentiable on \((a,b)\), then there exists \(c\in(a,b)\) such that \[ \left( f(b)-f(a)\right) g'(c) = \left( g(b)-g(a)\right) f'(c)\] If \(g'(c)\) is never zero on \((a,b)\), then \[ \frac{f'(c)}{g'(c)} = \frac{f(b)-f(a)}{g(b)-g(a)}\]

Theorem 5.3.6 (l'Hopital's Rule, 0/0 case). Assume \(f\) and \(g\) are continuous functions defined on an interval containing \(a\), and assume that \(f\) and \(g\) are differentiable on this interval. If \(f(a)=g(a)=0\), then \[ \lim_{x\to a} \frac{f'(x)}{g'(x)} = L \ \ \text{ implies } \ \ \lim_{x\to a} \frac{f(x)}{g(x)} = L\]

Chapter 6 Sequences and Series of Functions

Definition 6.2.1. For each \(n\in\N\), let \(f_n\) be a function with domain \(A\subset \R\). The sequence of functions \((f_n)\) converges pointwise to a function \(f:A\to\R\) iff for all \(x\in A\), the sequence \(f_n(x)\) converges to \(f(x)\).

Definition 6.2.3. For each \(n\in\N\), let \(f_n\) be a function with domain \(A\subset \R\). The sequence of functions \((f_n)\) converges uniformly to a function \(f:A\to\R\) iff for all \(\e>0\) there exists \(N\in\N\) such that \(n\ge N\) and \(x\in A\) implies that \(|f_n(x)-f(x)| < \e\).

Theorem 6.2.5 (Cauchy Criterion for Uniform Convergence). A sequence of functions \( (f_n)\) defined on a set \( A \subset \R\) converges uniformly on \(A\) iff for every \( \e>0\) there exists \( N\in\N\) such that \( |f_n(x)-f_m(x)|<\e\) whenever \(m,n\ge N\) and \(x\in A\).

Theorem 6.2.6. Let \((f_n)\) be a sequence of functions defined on \(A\subset\R\) that converges uniformly on \(A\) to a function \(f\). If each \(f_n\) is continuous at \(c\in A\), then \(f\) is continuous at \(c\).

Theorem 6.3.1. Suppose \((f_n)\) converges to \(f\) pointwise on the closed interval \([a,b]\) and that each \(f_n\) is differentiable on \([a,b]\). If \((f_n')\) converges uniformly on \([a,b]\) to \(g\), then \(f\) is differentiable and \(f'=g\).

Definition 6.4.1. For each \(n\in\N\), let \(f_n\) be a function defined on \(A\subset \R\). The infinite series \(\dsum_{n=1}^\infty f_n(x)\) converges pointwise to a function \(f:A\to\R\) iff the sequence of partial sums \[ s_k(x) = f_1(x) + \cdots + f_k(x) \] converges pointwise to \(f\). The series converges uniformly on \(A\) to \(f\) iff \((s_k(x))\) converges uniformly on \(A\) to \(f(x)\).

Theorem 6.4.2. Let each \(f_n\) be continuous on \(A\) and assume \(\dsum f_n\) converges uniformly on \(A\) to \(f\). Then \(f\) is continuous on \(A\).

Theorem 6.4.3. Suppose \(\dsum f_n\) converges pointwise on \(A\) to \(f\).
If \(\dsum f_n'\) converges uniformly to \(g\) on \(A\), then f is differentiable and \(f'=g\).

That is, if \(f(x)=\dsum f_n(x)\) and \(\dsum f_n'(x)\) converges uniformly, then \(f'(x) = \dsum f_n'(x)\).

Theorem 6.4.4 (Cauchy Criterion). A series \(\dsum f_n\) converges uniformly on \(A\subset\R\) iff for all \(\e>0\) there exists \(N\in\N\) such that \(n>m\ge N\) implies \[ | s_n(x)-s_m(x)| = | f_{m+1}(x) + f_{m+2}(x) + \cdots + f_n(x)| < \e \text{ for all } x\in A\]

Corollary 6.4.5 (Weierstrass M-Test). For each \(n\in\N\), let \(f_n\) be a function defined on \(A\subset\R\), and let \(M_n\in\R\) be positive such that \[ |f_n(x)| \le M_n \text{ for all } x\in A\] If \(\dsum M_n\) converges, then \(\dsum f_n\) converges uniformly on \(A\).

Theorem 6.5.1. If a power series \(\dsum_{n=0}^\infty a_n x^n\) converges at \(x_0\in\R\), then it converges absolutely for any \(x\) such that \(|x|<|x_0|\).

Theorem 6.5.2. If \(\dsum a_n x^n\) converges absolutely at \(x_0\), then the series converges uniformly on \([ -|x_0|, |x_0|]\).

Theorem 6.5.6. If \(\dsum a_n x^n\) converges on \((-R,R)\), then \(\dsum na_n x^{n-1}\) converges on \((-R,R)\).
Thus, \(\dsum na_n x^{n-1}\) is a power series that converges uniformly on any compact subset of \((-R,R)\).

Theorem 6.5.7. Assume that \(f(x)=\dsum a_n x^n\) converges on an interval \(A\subset\R\).
Then \(f\) is continuous on \(A\), differentiable on any interval \((-R,R)\subset A\), and \[ f'(x)=\sum na_n x^{n-1}\] Furthermore, \(f\) is infinitely differentiable where the derivatives can be taken term by term.

Theorem 6.6.3 (Lagrange's Remainder Theorem). Let \(f\) be differentiable \(N+1\) times on \( (-R,R)\), define \[ a_n = \frac{f^{(n)}(0)}{n!} \hspace{2em} \text{ for }\hspace{0.5em} n=0, 1, \ldots, N \] and let \[ S_N(x)=a_0 + a_1 x + a_2 x^2 + \cdots + a_Nx^N\] Given \( x\ne 0\) in \( (-R,R)\), there exists a point \(c\) satisfying \( |c|<|x|\) where the error function \[ E_N(x) = f(x) - S_N(x) \hspace{2em} \text{ satisfies } \hspace{2em} E_N(x)=\frac{f^{(N+1)}(c)}{(N+1)!} x^{N+1} \]

Theorem 6.7.1 (Weierstrass Approximation Theorem). Let \(f:[a,b]\to \R\) be continuous.
Given \( \e>0\) there exists a polynomial \(p(x)\) satisfying \[ | f(x) - p(x) | < \e \] for all \( x\in[a,b]\).

Chapter 7 The Riemann Integral

Definition 7.2.1. A partition \(P\) of \([a,b]\) is a finite ordered set \[ P = \{ a=x_0 < x_1 < \cdots < x_n = b\}\] Assume \(f\) is bounded on \( [a,b]\). For each subinterval of \([x_{k-1},x_k]\) of \(P\) let \[ m_k = \inf\{f(x) \ | \ x\in [x_{k-1},x_k]\} \ \ \text{ and }\ \ M_k = \sup\{f(x) \ | \ x\in [x_{k-1},x_k]\} \] The lower sum of \(f\) with respect to \(P\) is given by \[ L(f,P) = \sum_{k=1}^n m_k (x_k-x_{k-1} )\] and the upper sum of \(f\) with respect to \(P\) is given by \[ U(f,P) \sum_{k=1}^n M_k (x_k-x_{k-1} )\]

Definition 7.2.2. A partition \(Q\) is a refinement of a partition \(P\) if \(Q\) contains all of the points of \(P\). We denote this by \(P\subset Q\).

Lemma 7.2.3. If \(P\subset Q\), then \(L(f,P) \le L(f,Q)\) and \(U(f,Q)\le U(f,P)\).

Lemma 7.2.4. If \(P_1\) and \(P_2\) are any two partitions of \([a,b]\), then \( L(f,P_1) \le U(f,P_2) \).

Definition 7.2.5. Let \(\mathcal{P}\) be the set of all partitions of the interval \([a,b]\).

Define the upper integral of \(f\) to be \[ U(f)= \inf \{U(f,P)\ |\ P\in\mathcal{P}\} \]
Define the lower integral of \(f\) to be \[ L(f)= \sup \{L(f,P)\ |\ P\in\mathcal{P}\} \]

Lemma 7.2.6. If \(f\) is bounded on \([a,b]\), then \( U(f) \ge L(f)\).

Definition 7.2.7: A bounded function \(f\) defined on \([a,b]\) is Riemann-integrable iff \(U(f)=L(f)\). We define \[\int_a^b f = U(f) = L(f)\]

Theorem 7.2.8: A bounded function \(f\) is integrable on \([a,b]\) iff for all \(\e>0\) there exists a partition \(P_\e\) of \([a,b]\) such that \[ U(f,P_\e) - L(f,P_\e) < \e\]

Theorem 7.2.9: If \(f\) is continuous on \([a,b]\), then \(f\) is integrable.

Theorem 7.4.4. Assume \( f_n\to f\) uniformly on on \([a,b]\) and that each \(f_n\) is integrable. Then \(f\) is integrable and \[ \lim_{n\to\infty} \int_a^b f_n = \int_a^b f \]

Theorem 7.5.1 (Fundamental Theorem of Calculus).

1. If \(f:[a,b]\to\R\) is integrable and \(F:[a,b]\to\R\) satisfies \(F'(x)=f(x)\) for all \(x\in[a,b]\), then \[ \int_a^b f = F(b)-F(a) \]
2. Let \(g:[a,b]\to\R\) be integrable, and for \(x\in[a,b]\) define \[ G(x)=\int_a^x g\] Then \(G\) is continuous on \( [a,b]\).
   If \(g\) is continuous at some point \(c\in[a,b]\) then \(G\) is differentiable at \(c\) and \(G'(c)=g(c)\).