(Solved): Calculus Let {an} be a sequence. Definition. subsequence of {an} is a sequence made up of ...

Calculus Let $\left\{a_n\right\}$ be a sequence. Definition. $?$ subsequence of $\left\{a_n\right\}$ is a sequence made up of some of the terms of $\left\{a_n\right\}$ in increasing index order. For example, $a_1,a_3,a_5,\dots$ is a subsequence but $a_3,a_1,a_7,a_5,\dots$ is not (because the subscripts are not in increasing order). Here is an important result about subsequences: Proposition. Suppose $\left\{a_n\right\}$ is a sequence and $\left\{b_k\right\}$ is a subsequence of $\left\{a_n\right\}$. If $\left\{a_n\right\}$ converges to $L$, say, then $\left\{b_k\right\}$ converges to $L$ also. (a) Write down a convergent subsequence of the sequence $1,1,2,\frac{1}{2},3,\frac{1}{3},4,\frac{1}{4},\dots$ (b) Use the Monotone Convergence Theorem and the Proposition above to prove the following result (make sure you reference where you are using each theorem): Proposition. Let $\left\{a_n\right\}$ be a bounded sequence, and suppose that there is a subsequence of $\left\{a_n\right\}$ which converges to 0 and another subsequence of $\left\{a_n\right\}$ which converges to 1 . Prove that $\left\{a_n\right\}$ is not monotone. [Hint: Try a proof by contradiction (i.e. suppose that $\left\{a_n\right\}$ is monotone. What would happen?)]