# Proofs - Analysis I MT22 > Source: https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/proofs/ · Updated: 2023-01-03 · Tags: proofs, analysis, mt22 ### Basic properties Prove the triangle inequality for the reals: > If $x, y \in \mathbb{R}$ then $|x+y| \le |x|+|y|$. :: Todo? Prove the *reverse* triangle inequality for reals: > If $x, y \in \mathbb{R}$ then $\big||x| - |y|\big| \le |x - y|$. :: Todo? ### Sequences Prove the squeeze/sandwich theorem: > Assume $a_n \le x_n \le b_n$ and that $a_n \to L$ and $b_n \to L$. Then $x_n \to L$. :: Todo? Prove the preservation of weak inequalities for sequences: > Assume $a_n \le b_n$ and that $a_n \to L$ and $b_n \to M$. Then $L \le M$. :: Todo? Prove the quotient law in algebra of limits: > Assume $a_n \to a$ and $b_n \to b$. Then $\frac{a_n}{b_n} \to \frac{a}{b}$ :: Todo? Prove that if $(a_n)$ is a convergent sequence, then $(a_n)$ is bounded. :: Todo? Prove that limits are unique. :: Todo? Prove that if $(a_n)$ is monotically increasing, then either $a_n \to \infty$ or $a_n \to \sup(\\{a_n : n \in \mathbb{N}\\})$. :: Todo? Prove that if $(a_n)$ is monotically decreasing, then either $a_n \to -\infty$ or $a_n \to \inf(\\{a_n : n \in \mathbb{N}\\})$. :: Todo? Let $A\subseteq\mathbb{R}$. Prove $\sup(A)=\alpha$ if and only if: - $\alpha$ is an upper bound of $A$, and - Given any $\epsilon > 0$, there exists some $x \in A$ such that $\alpha - \epsilon < x$. :: Todo? Let $A \subseteq \mathbb{R}$. Prove $\inf(A) = \alpha$ if and only if: - $\alpha$ is a lower bound of $A$, and - Given any $\epsilon > 0$, there exists some $x \in A$ such that $x < \alpha + \epsilon$. :: Todo? Prove that every sequence contains a monotone subsequence (scenic viewpoint theorem). :: Todo? Prove, the Bolzano-Weierstrass Theorem: > Every bounded sequence has a convergent subsequence. Hint: Appeal to two smaller theorems. :: Todo? Prove that if $(a_n)$ is a Cauchy sequence, then $(a_n)$ is bounded. :: Todo? Prove the Cauchy criterion for convergence: > A sequence converges if and only if it is Cauchy. :: Todo? ### Series Prove the validity of the alternating series test: > If $(a_k)$ is a monotonically decreasing sequence and $a_k \to 0$, then $\sum^\infty_{k=1}(-1)^{k+1}a_k$ converges. :: Todo? Prove the limit form of the comparison test: > Let $(a_k)$ and $(b_k)$ be real sequences of positive terms, and assume that there is $L \in (0, \infty)$ such that $\frac{a_k}{b_k} \to L$ as $k \to \infty$. Then $\sum a_k$ converges if and only if $\sum b_k$ converges. :: - First prove the normal form of the comparison test, which follows from $a_k$ being a monotone increasing sequence - Then consider what $\frac{a_k}{b_k} \to L$ tells you by taking $\varepsilon = L/2$ and use the regular comparison test for both directions. (analysis i, page 90) Prove the alternating series test: > Let $(u_k)$ be a real sequence, and consider the series $\sum^\infty_{k=1} (-1)^{k-1} u_k$. If $u_k \ge 0$ for $k \ge 1$, $(u_k)$ is decreasing, and $u_k \to 0$ as $k \to \infty$, then the series converges. :: - Group terms together in different ways in order to see that both the $s_{2n}$ and $s_{2n+1}$ series of partial sums are convergent by the monotone sequence theorem. (analysis i, page 88) Prove the ratio test: > Let $(a_k)$ be a real sequence of positive terms. Assume that $\frac{a_{k+1}\\,}{a_k}$ converges as $k \to \infty$, say to a limit $L$. Then if $0 \le L < 1$, then $\sum a_k$ converges and if $L > 1$, then $\sum a_k$ diverges. :: Need to handle two cases seperately, but in both we compare with a geometric series. In the case where $0 \le L < 1$, this is $\alpha = \frac{1+L}{2}$ with $\varepsilon = \alpha - L$, in the other case it’s $\alpha = 2$. (analysis i, page 92) Prove the integral test: > Let $f : [1, \infty) \to \mathbb R$ be a function. Assume that > - $f$ is nonnegative > - $f$ is decreasing > - $\int^{k+1}_k f(x) \text d x$ exists for each $k \ge 1$ > Then let $s_n = \sum^n_{k=1} f(k)$ and $I_n = \int^n_1 f(x) \text d x$. Then > - $\sigma_n = s_n - I_n$ converges and $0 \le \lim_{n \to \infty} \sigma_n \le f(1)$ > - $s_n$ converges iff $I_n$ converegs. :: - Show that $\sigma_n$ is bounded below and decreasing, by adding up all the inequalities like $f(n) \le \int^n_{n-1} f(x) \text d x \le f(n-1)$ to get $0 \le \sigma_n \le f(1)$ and then mess with $\sigma_{n+1} - \sigma_n$ to show it’s negative. - Then the other result follows from the AOL. (analysis i, page 96). ### Radius of convergence Prove that power series converge within their radius of convergence $R$: > Let $\sum c_k z^k$ be a power series with radius of convergence $R$. Then if $R > 0$ and $|z| < R$ then $\sum c_k z^k$ converges absolutely and hence converges. If $|z| > R$ then $\sum c_k z^k$ diverges. :: - Two cases parts of the proof, absolute convergence within radius of convergence and then divergence outside it - Absolute convergence within radius of convergence: There is some $S$ less than the radius of convergence, and let $\varepsilon = R - S$ to find a $\rho$ less than or equal to $R$ where the series converges absolutely. Then use the comparison test. - Divergence outside radius of convergence: Use contradiction. Can bound the terms being summed by $M$, and then can find some $\rho$ where $R < \rho < |z|$ where $0 \le |c_k \rho^k \le |c_k z^k| \left| \frac \rho z \right|^k \le M \left |\frac \rho z\right|^k$ which converges as it’s a geometric series with common ratio less than $1$. --- Olly Britton — https://ollybritton.com. Machine-readable index: https://ollybritton.com/llms.txt