Due: October 20th, 2017
Math 142A Assignment 3
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Let $(a_n)$ be a sequence of real numbers.
Determine whether each statement below implies that $(a_n)$ converges.
Then determine whether each statement below is implied by the assumption that $(a_n)$ converges.
 There is some $a\in\mathbb{R}$ so that for every $\epsilon > 0$ there is some $N \in \mathbb{N}$ so that for every $n \geq N$, $a_na < 10\epsilon + \epsilon^2$.
 For every $\epsilon > 0$ there is some $N \in \mathbb{N}$ so that for every $n > N$, $a_{n+1}a_n < \epsilon$.
 There is some $a \in \mathbb{R}$ and some $N \in \mathbb{N}$ so that for every $\epsilon > 0$ and every $n > N$, $a_na < \epsilon$.
 The sequence $\left(a_n\right)$ is monotonic and bounded.

A set of real numbers $S \subseteq \mathbb{R}$ is said to be open if for every $x \in S$ there is some $\epsilon > 0$ such that $(x\epsilon, x+\epsilon) \subset S$.
 Show that for any real numbers $a < b$, the interval $(a, b)$ is open.
 Show that if $G$ is open, then $\mathbb{R}\setminus G$ is closed.
 Show that if $F$ is closed, then $\mathbb{R}\setminus F$ is open.

The Nested Interval Theorem
Suppose that $(a_n)$ and $(b_n)$ are sequences such that for every $n$, $a_{n} \leq a_{n+1} \leq b_{n+1} \leq b_n$, so that $[a_{n+1}, b_{n+1}] \subseteq [a_n, b_n]$.
Prove that the set \[S = \bigcap_{n=1}^{\infty} [a_n, b_n] = \left\{x \in \mathbb{R} : x \in [a_n, b_n]\,\forall n\right\}\] is nonempty.

Let $(b_n)$ be a bounded sequence of nonnegative numbers.
Show that for any $r \in [0, 1)$, the sequence of partial sums $(s_n)$ given by \[s_n = \sum_{k=1}^n b_kr^k\] converges.


The Squeeze Theorem (for Sequences)
Show that if $(a_n), (b_n),$ and $(c_n)$ are sequences such that $(a_n)$ and $(c_n)$ converge to $L$, and for some $N \in \mathbb{N}$ and every $n > N$ we have $a_n \leq b_n \leq c_n$, then $(b_n)$ converges to $L$.

Show that if $(a_n)$ and $(b_n)$ are sequences such that $(a_n)$ diverges to infinity and for some $N \in \mathbb{N}$, every $n > N$, and some $c > 0$ we have $b_n > ca_n$, then $(b_n)$ diverges to infinity.

Bonus:
Suppose $(a_n)_n$ is a sequence with the property that every subsequence $(a_{n_k})_k$ has a further subsequence $(a_{n_{k_j}})_j$ which converges to $a$.
Does $(a_n)_n$ converge to $a$?
Prove it or provide a counterexample.