Connected and Locally Path-Connected Implies Path Connected

Theorem
Let $X$ be a connected and locally path-connected topological space.

Then $X$ is path-connected.

Proof
By:
 * Path Components are Closed
 * Path Components of Locally Path-Connected Space are Open

the path components of $X$ are clopen.

Because $X$ is connected, every path component equals $X$.

That is, $X$ is path-connected.