If $\displaystyle dimM=m<p$, show that every map(maybe continuous) $\displaystyle M^m \to S^p$ is homotopic to a constant.

This is the problem 5 in chap8. of 'topology from the differentiable viewpoint(Milnor)'.

I proved it when the map is not onto. But I think it can be onto.

Please help me.