I'm trying to understand intuitively the proposition that:
Any loop $p:[0,1] \to S^1$ is end point preserving homotopic to a loop which doesn't change direction.
Surely a loop round a circle starting at one point and ending at the same point is very restrictive in how it travels round the circle - it either goes round or it doesn't.
What are some examples of how a loop can go around a circle?.. I.e. can the 'loop' stop half way, go back a bit, then forward until the end?
Thanks!