Question

Prove that a sequence $\left\{a_k\right\}$that is both increasing and decreasing is constant.

Short answer

Proved

Step-by-step solution

Step 1. Given

Consider the sequence$\left\{a_k\right\}$

Step 2. Prove

$$\begin{gathered} \mathrm{According}\mathrm{to}\mathrm{the}\mathrm{definition}\mathrm{the}\mathrm{sequence}\left\{{\mathrm{a}}_{\mathrm{k}}\right\}\mathrm{is}\mathrm{an}\mathrm{increasing}\mathrm{sequence}\mathrm{if}{a}_{k+1}\ge a_k,fork\ge 1 \\ \mathrm{Since},\mathrm{the}\mathrm{given}\mathrm{sequence}\mathrm{is}\mathrm{increasing}\mathrm{sequence},\mathrm{hence},a_{k+1}\ge a_k,......\left(1\right)forallk\ge 1. \\ \mathrm{Also},\mathrm{according}\mathrm{to}\mathrm{the}\mathrm{definition}\mathrm{the}\mathrm{sequence}\left\{\mathrm{a}\right\}\mathrm{is}\mathrm{a}\mathrm{decreasing}\mathrm{sequence}\mathrm{if}{a}_{k+1}\le a_k,fork\ge 1 \\ \mathrm{Since},\mathrm{the}\mathrm{given}\mathrm{sequence}\mathrm{is}\mathrm{increasing}\mathrm{sequence},\mathrm{hence},,a_{k+1}\ge a_k,......\left(1\right)forallk\ge 1. \end{gathered}$$