(algebraic topology) question about the cellular approximation theorem

Since $n<k$, $n$-th skeleton of $S^k$ is just a point, but the $n$-th skeleton of $S^n$ is already $S^n$ itself. Cellular approximation implies that $f$ is homotopic to $g: S^n \to S^k$ with the property $$g(S^n)=g(\text{sk}^n(S^n)) \subset \text{sk}^n(S^k)=\{\text{pt} \},$$ so $g$ is a map to a point.

Since $f$ is homotopic to some cellular map $g$ we only need to consider $g$ for now. We have that $g$ maps the $j$-skeleton of $S^n$, $0 \leq j \leq n$, to the $j$-skeleton of $S^k$. As $n < k$ (and we have these CW-structures), you can only map to the point that gives the $0$-skeleton of $S^k$ as we first add new cells at level $k$.