Continuous deformation of loop to point. Suppose I have a homotopy from a loop around the origin to a constant loop which is not the origin.
Prove that the origin is in the image of the homotopy.
Basically prove that if I deform a loop to a point, at some point in time it has to cross the origin.
I have tried to prove this but I get stuck trying to express mathematically that the loop can't break.

If the origin is not in the image, then consider your original loop as a loop in space $\mathbf{R}^2 \backslash(0,0)$.
Since your deformation avoids the origin, it can be viewed, again, as a deformation in the "punctured" space that deforms the loop encompassing the origin to the constant map.
But the loop in the constant map is not null-homotopic since the punctured space is equal to the circle, and the loop is nontrivial (homotopically speaking of course.)
[You may need to put more flesh on these!]
