I'm hoping someone can check my proof of the following problem. I feel like 'check my proof' questions are sort of suboptimal, but as I'm purely a self-studier and in particular the book I'm working from doesn't have solutions, this is sort of my only means of verification, which is always nice to have. So the help is always much appreciated!
"Let $X$ be a space and $x$ a point at which $X$ is locally compact. Prove that there is a local basis $\mathcal{B}$ at $x$ such that $\bar{B}$ is compact for each $B \in \mathcal{B}$."
Answer given below is the original, and a correct, proof.