Question

Prove that every convergent sequence is bounded.

Short answer

Proved that every convergent sequence is bounded

Step-by-step solution

Step 1. Given information

Let consider the given convergent sequence$\left\{a_k\right\}$ converging to limitL.

Step 2. Prove that the convergent sequence is bounded

The strategy is to prove that the sequence is bounded,show that there exits a positive integer Msuch that $\left|a_k\right|\le Mforallk.$

The sequence $\left\{a_k\right\}$is convergent and converges to L

By the defination of convergence,for $\epsilon =1(>0)$ there is a positive integer N,such that

$$\begin{gathered} \left|a_k-L\right|<\epsilon fork\ge N \\ \left|a_k-L\right|<1fork\ge N \end{gathered}$$

for all $k\ge N,$fix Nsuch that

$$\begin{gathered} \left|a_k\right|\le \left|a_k-L\right|+\left|L\right| \\ \left|a_k\right|<1+\left|L\right|(Because\left|a_k-L\right|<1) \end{gathered}$$

For all $k\ge N$,choose $M=max\left\{\left|a_1\right|,\left|a_2\right|,\left|a_3\right|,............\left|a_N\right|,1+\left|L\right|\right\}$

Therefore,$\left|a_k\right|<Mfork\ge M$

Thus the sequence$\left\{a_k\right\}$ is bounded

Therefore,every convergent sequence is bounded