Homework 6: More Limits

Instructions. Assignments should be stapled and written clearly and legibly. Problems 6 and 7 are optional.

1. Let \((a_n)\) be a convergent sequence. Suppose that \(\lim a_n \gt 0\). Use the definition of a limit to prove that there exists \(N\in\mathbb{N}\) such that \(a_n \gt 0\) for all \(n\geq N\).
2. From any given sequence \((a_n)\) we can form the related sequence \((b_n)=(5a_n+2)\). Use the definition of convergence of a sequence to prove that if \((a_n)\) converges to \(20\), then \((b_n)\) converges to . (First fill in the blank.)
3. Let \((a_n)\) and \((b_n)\) be sequences. Suppose that \((a_n)\) converges to 0.
   1. Using the definition of convergence, prove that if \((b_n)\) is bounded, then the sequence \((a_n b_n)\) converges. (Note that you may not assume that \((b_n)\) converges.)
   2. If the sequence \((b_n)\) is not bounded, must the sequence \((a_n b_n)\) necessarily converge? If so, prove it. If not, give a counterexample.
4. Give an example of a sequence \((a_n)\) such that
   1. \((a_n)\) converges to 0, but \(a_n\neq 0\) for all \(n\).
   2. \((a_n)\) is bounded but does not converge.
   3. \(\lim\limits_{n\to\infty} \dfrac{a_{n+1}}{a_n} = 1\) but \((a_n)\) does not converge.
   4. \((|a_n|)\) converges but \((a_n)\) does not.
5. Give examples of the following:
   1. two divergent sequences \((a_n)\) and \((b_n)\) for which \((a_n+b_n)\) coverges.
   2. two divergent sequences \((c_n)\) and \((d_n)\) for which \((c_nd_n)\) converges.
6. (Challenging) Suppose that \((a_n)\) is a convergent sequence and \(f:\mathbb{N}\to \mathbb{N}\) is a bijection. Determine whether \((a_{f(n)})\) converges. Prove your answer.
7. (More Challenging)
   1. Prove that if a sequence \((a_n)\) is convergent, then the sequence of averages \begin{equation*} b_n=\frac{a_1+a_2+\cdots + a_n}{n} \end{equation*} is also convergent, and converges to the same limit.
   2. Show by example that it is possible for a sequence to diverge but its sequence of averages to converge.