Cauchy sequence manipulation

Question

Let $(x_k)_k\subseteq\mathbb{R}$ be a Cauchy sequence. Is $(e^{1/x_k})_k$ a Cauchy sequence as well? I am not sure how to approach this question.

Answer 1

Since you are in $\mathbb{R}$, a sequence converges iff the sequence is Cauchy. So since $(x_k)_k$ is Cauchy, $(x_k)_k$ converges to some $x\in\mathbb{R}$.

• If $x\neq 0$, by continuity, $(e^{1/x_k})_k$ converges to $e^{1/x}$, so that $(e^{1/x_k})_k$ is Cauchy.
• If $x=0$, $1/x_k\,\rightarrow_k\infty$, so there exists $N\in \mathbb{N}$ and $\varepsilon >0$ such that for all $k\geq N$, $$|x_{k+1}-x_k|\geq \varepsilon,$$ so that $(1/x_k)_k$ is not Cauchy and therefore neither $(e^{1/x_k})_k$.