Locally Euclidean Space is Locally Compact/Proof 2

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Theorem

Let $M$ be a locally Euclidean space of some dimension $d$.

Then $M$ is locally compact.

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.

$\blacksquare$