a connected and locally path connected space is path connected

Theorem. A connected, locally path connected topological space is path connected.

Proof. Let $X$ be the space and fix $p\in X$. Let $C$ be the set of all points in$X$ that can be joined to $p$ by a path. $C$ is nonempty so it is enough to show that $C$ is both closed and open.

To show first that $C$ is open: Let $c$ be in $C$ and choose an open path connected neighborhood $U$ of $c$. If $u\in U$ we can find a path joining $u$ to $c$ and then join that path to a path from $p$ to $c$. Hence $u$ is in $C$.

To show that $C$ is closed: Let $c$ be in $\overline{C}$ and choose an open path connected neighborhood $U$ of $c$. Then $C\cap U\ne \mathrm{\varnothing }$. Choose $q\in C\cap U$. Then $c$ can be joined to $q$ by a path and $q$ can be joined to $p$ by a path, so by addition of paths, $p$ can be joined to $c$ by a path, that is, $c\in C$.