Locally compact Hausdorff implies completely regular

This article gives the statement and possibly, proof, of an implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property (i.e., locally compact Hausdorff space) must also satisfy the second topological space property (i.e., completely regular space)
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Statement

Any locally compact Hausdorff space is completely regular.

Related facts

• Compact Hausdorff implies normal
• Paracompact Hausdorff implies normal

Facts used

1. Any locally compact Hausdorff space possesses a one-point compactification; hence, it is a subset (with the subspace topology) of a compact Hausdorff space.
2. Compact Hausdorff implies normal
3. Normal implies completely regular
4. Complete regularity is hereditary

Proof

Given: A locally compact Hausdorff space $X$.

To prove: $X$ is a completely regular space.

Proof:

1. $X$ is a subspace of a compact Hausdorff space $Y$: This follows from fact (1); we can take $Y$ to be the one-point compactification of $X$.
2. $Y$ is a normal space: This follows from fact (2).
3. $Y$ is a completely regular space: This follows from the previous step and fact (3).
4. $X$ is a completely regular space: This follows from the previous step and fact (4).