Due: October 20th, 2017

Math 142A Assignment 3

Answers to these problems are now available here. The $\LaTeX$ file used to produce them is here.

  1. 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.
    1. There is some $a\in\mathbb{R}$ so that for every $\epsilon > 0$ there is some $N \in \mathbb{N}$ so that for every $n \geq N$, $|a_n-a| < 10\epsilon + \epsilon^2$.
    2. For every $\epsilon > 0$ there is some $N \in \mathbb{N}$ so that for every $n > N$, $|a_{n+1}-a_n| < \epsilon$.
    3. There is some $a \in \mathbb{R}$ and some $N \in \mathbb{N}$ so that for every $\epsilon > 0$ and every $n > N$, $|a_n-a| < \epsilon$.
    4. The sequence $\left(|a_n|\right)$ is monotonic and bounded.
  2. 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 $(x-\epsilon, x+\epsilon) \subset S$.
    1. Show that for any real numbers $a < b$, the interval $(a, b)$ is open.
    2. Show that if $G$ is open, then $\mathbb{R}\setminus G$ is closed.
    3. Show that if $F$ is closed, then $\mathbb{R}\setminus F$ is open.
  3. 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 non-empty.
  4. Let $(b_n)$ be a bounded sequence of non-negative 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.
    1. 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 $L$.
    2. 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.
  5. 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.