A locally constant sheaf on a contractible space is constant.

Question

This is Exercise II.4. in the book [Sheaves on manifolds, Kashiwara and Schapira].

If X is simply connected and locally path connected, it is known that every locally constant sheaf F on X is constant.

Now X is contractible (thus simply-connected and path connected of course, but not necessarily locally path connected). How to prove that a locally constant sheaf F on X is still constant?