If $(a_n)_{n\in\mathbb{N}}$ is a Cauchy sequence, then $(a_n ^2)_{n\in\mathbb{N}}$ is also a Cauchy sequence

Question

If the sequence $\{a_n\}$ with $n∈\mathbb{N}$ is a Cauchy sequence, then $\{a_n^2\}_{n∈\mathbb{N}}$ is also a Cauchy sequence. How do we prove it?

Answer 1

Directly from definition:

Take any $\;\epsilon>0\;$ . Since $\;\{a_n\}_{n\in\Bbb N}\;$ is a Cauchy sequence it is bounded, there exists $\;M\in\Bbb R\;$ s.t. for any $\;k\in\Bbb N\;,\;\;|a_k|\le M\;$ , and since it is also a Cauchy sequence there exists $\;N\in\Bbb N\;$ s.t. for all $\;n,m>N\;,\;\;|a_n-a_m|<\frac\epsilon{2M}\;$ , thus:

$$\forall\,n,m>N\;,\;\;|a_n^2-a_m^2|=|a_n-a_m|\,|a_n+a_m|\le\frac\epsilon{2M}\left(|a_n|+|a_m|\right)=\epsilon$$

and $\;\{a_n^2\}_{n\in\Bbb N}\;\;$ is Cauchy.