I've read this online, but I haven't seen this proved in Munkres Topology. Has it been? If so, where?
In any case, here is my attempt to show it using the tools given in the book. Please verify.
The book has 2 exercises on contractible spaces. One says contractible spaces are path connected and the other says a space is contractible if and only if the space has the same homotopy type of a one-point space. Since simply connected is defined as path connected and trivial fundamental group, I think that if a space has the same homotopy type as that of a one-point space, then the fundamental group of the space is trivial. Is that right? If so, I can work out the details myself.
Exercise 51.3b
Exercise 58.5
Definition of fundamental group
Definition of simply connected
The context is the claim made here in Example 70.1
Best Answer
I thought of one. Use Corollary 58.6 with $h$ as the identity map.
Since $h$ is nullhomotopic by definition of contractible, the induced homomorphism $h_{*}$ is both the identity isomorphism and the trivial homomorphism.
This means all loop classes are what they are mapped to, which is the identity class!