Proving an "inevitable intersection"

Question

So, I need to prove that if a curve $C$ is homotopic to a point (with homotopy $H$ where deformation happens exclusively in the same number of dimensions as $C$), then all of the points within that curve are intersected by the curve created by $H$ for some value of $t \in [0,1]$. So far, I have tried to prove this by demonstrating that the function that maps the intersection of all curves $H(x,t)$ to any plane within the curves is continuous, but so far said approach has not helped. Could you please tell me how to prove this?