Prove $a_n:=\exp\left(-\sqrt{\frac{1}{n}+|x|^2}\right)$ is a Cauchy sequence.

Question

Let the space be $L^2$ and equipped with the usual norm. Prove $a_n:=\exp\left(-\sqrt{\frac{1}{n}+|x|^2}\right)$ is a Cauchy sequence in $L^2$.

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$0< a_n <1$ and monotone increasing. So for a given $x$ we can find $N$ such that $n,m\geq N$ means that $a_n,a_m\in (e^{-|x|}-\epsilon,e^{-|x|}+\epsilon )$ with $\epsilon$ arbitrary, and the norm
$$ \sqrt {\int |a_n-a_m|^2\;dx} $$ goes to zero as $a_n\to a_m$ in absolute value.

But this is pointwise convergence. Anyone know if uniform convergence can be proved?

EDIT: Another question. Say a space is complete. Then every Cauchy sequence is convergent in the space. But is it always pointwise or uniform convergent or does it depend on the metric of the space? And if it depends on the metric, how does it depend on the metric?

Answer 1

You ask about uniform convergence, but note uniform convergence does not imply convergence in $L^2(\mathbb R).$

Hints 1. The given functions are even so we need only prove convergence in $L^2([0,\infty)).$

2. Note $(e^{-\sqrt {x^2+1/n}} - e^{-x})^2 \le (2e^{-x})^2.$ This sets you up for the DCT.