Locally Euclidean Space is Locally Compact/Proof 2

Proof
Let $m \in M$ be arbitrary.

From Locally Euclidean Space has Countable Neighborhood Basis Homeomorphic to Closed Balls:
 * there exists a countable neighborhood basis $\family{K_n}_{n \in \N}$ of $m$ where each $N_n$ is the homeomorphic image of a closed ball of $\R^d$

From Closed Ball in Euclidean Space is Compact:
 * $\forall n \in \N: K_n$ is the homeomorphic image of a compact subspace

From Continuous Image of Compact Space is Compact:
 * $\forall n \in \N: K_n$ is a compact subspace

Hence:
 * there exists a countable neighborhood basis $\family{K_n}_{n \in \N}$ of $m$ consisting of compact subspaces

It follows that $M$ is locally compact by definition.