This can be proved using the Brouwer degree. Smooth maps $f,g\colon\thinspace S^n\to S^n$ are homotopic if and only if they have the same Brouwer degree, and any continuous map is homotopic to a smooth one.

On second thoughts, this might be identical to Chris Gerig's answer. Milnor's lovely book "Topology from the Differentiable Viewpoint" contains the details.