Example of a locally compact metric space whose completion is not locally compact

Question

Can someone suggest an example of a locally compact metric space whose completion is not locally compact?

Answer 1

here it is one: Put your self in $l^{\infty}(\mathbb{N})$.For every $m \in \mathbb{N}$ consider the family $I_m=\{\frac{1}{m}(1+\frac{1}{i})(e_n), i,n \in \mathbb{N}\}$. Consider $X=\bigcup_{m \in \mathbb{N}}I_m$. Now $\frac{1}{m} e_i \in \bar{X}, \forall m,i \in \mathbb{N}$. So $0 \in \bar{X}$. But now in every ball centered in 0, there's the sequence $\{\frac{1}{m}e_i\}_{i \in \mathbb{N}}$ for $m$ big enough, and this sequence is of points with pairwise distance greater than a costant, so no Cauchy subsequence. And of course $X$ is locally compact being locally discrete(if this sounds to you too degenerate, observe that there's enough freedom to subsitute points in the definition of $I_m$ with any compact small enough shape centered in $\frac{1}{m}(1+\frac{1}{i}(e_n)$).