Proofs - Analysis I MT22
Basic properties
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 $.
If $x, y \in \mathbb{R}$ then $ \vert x+y \vert \le \vert x \vert + \vert y \vert $.
Todo?
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 $.
If $x, y \in \mathbb{R}$ then $\big \vert \vert x \vert - \vert y \vert \big \vert \le \vert x - y \vert $.
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 $x _ n \to L$.
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 $L \le M$.
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Prove the quotient law in algebra of limits:
Assume $a _ n \to a$ and $b _ n \to b$. Then $\frac{a _ n}{b _ n} \to \frac{a}{b}$
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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).
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Prove, the Bolzano-Weierstrass Theorem:
Every bounded sequence has a convergent subsequence.
Hint: Appeal to two smaller theorems.
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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.
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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.
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Todo?
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.
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.
- 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 < \vert z \vert $ where $0 \le \vert c _ k \rho^k \le \vert c _ k z^k \vert \left \vert \frac \rho z \right \vert ^k \le M \left \vert \frac \rho z\right \vert ^k$ which converges as it’s a geometric series with common ratio less than $1$.