Munkres defines a simply connected space $X$ as:
A path connected space in which $\pi_1(X,x_0)$ is the trivial one-element group for some $x_0\in X$, and hence for every $x_0\in X$.
He then goes on to prove that in a simply connected space $X$, any two paths having the same initial and final points are homotopic.
Why is such a proof required? Doesn't the fact that $\pi_1(X,x_0)$ contains only the identity element $[e_{x_{0}}]$ already prove that any two paths starting and ending at $x_0$ will homotopic?
Thanks.
The fundamental group concerns loops, i.e. paths for which the final point and the initial one coincide. If you have two paths which are not loops, but such that they share the same initial and final points, that claim is true but it needs a proof, since it does not follow "directly" from the triviality of $\pi_1$.