Due: October 13th, 2017

Math 142A Assignment 2

Answers to these problems are now available here. In some case these are sketches rather than full proofs, but they should at least give an indication of how to prove the result.

  1. Determine which of the following sequences in $n$ converge. Find the limits of those which do.
    1. $\left(\frac{n}{n+1}\right)$
    2. $\left(\sqrt{n+1}-\sqrt{n}\right)$
    3. $\left(n\right)$
    4. $(3)$
    5. $\left(\frac{(-1)^n}{n^2}\right)$
  2. Prove each of the following statements that is true. Disprove each that is false.
    1. If the sequence $(a_n^2)$ converges, then $(a_n)$ also converges.
    2. If the sequence $(a_n + b_n)$ converges, then $(a_n)$ and $(b_n)$ converge.
    3. If $(a_n)$ converges to $a > 0$, then there is some $N \in \mathbb{N}$ such that $a_n > 0$ for every $n > N$.
  3. Suppose that $(a_n)$ and $(b_n)$ converge to $a$ and $b$ respectively. Let $(c_n)$ be the sequence defined by $c_{2k} = a_k$ and $c_{2k-1} = b_k$. Show that $(c_n)$ converges if and only if $a = b$.
  4. Let $(a_n)$ be a sequence. Show that $(a_n)$ converges to $a$ if and only if $(a-a_n)$ converges to $0$.
  5. Let $S$ be a non-empty set which is bounded above, with least upper bound $m$. Show that there is a sequence $(a_n)$ such that $a_n \in S$ and $\displaystyle\lim_{n\to\infty}a_n = m$.
  6. Bonus:

    Does there exist a sequence $(r_n)$ such that for each $\alpha \in \mathbb{R}$ there is some subsequence $(r_{n_k})$ which converges to $\alpha$? Prove it or disprove it.