= Solution
Triangulate $S^n$ and $S^m$. By the <Simplicial approximation theorem>, after subdividing the domain, a continuous map $f:S^n\to S^m$ is homotopic to the realization of a simplicial map $g$. Since $n<m$, the image of $|g|$ lies in the $n$-skeleton of the target triangulation and therefore misses an interior point of every $m$-simplex. A sphere with one point removed is homeomorphic to the <Euclidean space> $\mathbb R^m$, which is <contractible>. Hence $|g|$, and therefore $f$, is homotopic to a constant map. Thus
$$
\boxed{n<m\quad\Longrightarrow\quad
\text{every }f:S^n\to S^m\text{ is null-homotopic}.}
$$
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