I'm trying to figure out whether my proof is correct for a question I'm trying to tackle in Topology by James R. Munkres.
Task: Let $X$ be locally path connected. Show that every connected open set in $X$ is path connected.
My attempt at a proof: Well, every open subset of the locally path connected space $X$ is locally path connected. In addition, the path components and components of $X$ are the same in view of Theorem $25.5$ (p. $162$), which states that if a space $X$ is locally path connected, then the components and the path components of $X$ are the same. Altogether, this implies that then every connected open set in $X$ is path connected.
Am I right, or do I need to make any changes? Please provide your input, thanks.