Proofs - Analysis I MT22

Prove the triangle inequality for the reals:

If $x, y \in \mathbb{R}$ then $ \vert x+y \vert \le \vert x \vert + \vert y \vert $.

Prove the reverse triangle inequality for reals:

If $x, y \in \mathbb{R}$ then $\big \vert \vert x \vert - \vert y \vert \big \vert \le \vert x - y \vert $.

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 $x _ n \to L$.

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 $L \le M$.

Prove the quotient law in algebra of limits:

Assume $a _ n \to a$ and $b _ n \to b$. Then $\frac{a _ n}{b _ n} \to \frac{a}{b}$

Prove that if $(a _ n)$ is a convergent sequence, then $(a _ n)$ is bounded.

Prove that limits are unique.

Prove that if $(a _ n)$ is monotically increasing, then either $a _ n \to \infty$ or $a _ n \to \sup(\{a _ n : n \in \mathbb{N}\})$.

Prove that if $(a _ n)$ is monotically decreasing, then either $a _ n \to -\infty$ or $a _ n \to \inf(\{a _ n : n \in \mathbb{N}\})$.

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 $\alpha - \epsilon < x$.

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 $x < \alpha + \epsilon$.

Prove that every sequence contains a monotone subsequence (scenic viewpoint theorem).

Prove, the Bolzano-Weierstrass Theorem:

Every bounded sequence has a convergent subsequence.

Hint: Appeal to two smaller theorems.

Prove that if $(a _ n)$ is a Cauchy sequence, then $(a _ n)$ is bounded.

Prove the Cauchy criterion for convergence:

A sequence converges if and only if it is Cauchy.

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.

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.

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.

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.

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.

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 $ \vert z \vert < R$ then $\sum c _ k z^k$ converges absolutely and hence converges. If $ \vert z \vert > R$ then $\sum c _ k z^k$ diverges.