Each homotopy representative is supported on a finite CW subcomplex

The Euclidean sphere and disk are closed and bounded, as are their products with $I=[0,1]$, regarded as subsets of a finite-dimensional Euclidean space. Thus [F3] makes them compact. $D^0$ is a singleton and $S^0$ consists of two points, which are compact by taking one covering member for each of finitely many points. Their products with $I$ are a single interval or two intervals, also closed bounded Euclidean subsets after the usual embeddings.

The sphere has a finite $n$-dimensional CW structure with its designated basepoint as a vertex. One concrete construction for $n\ge 1$ attaches one $n$-disk to a point by collapsing its entire boundary. To identify the quotient with $S^n$, send $u\in D^n$ of norm $r>0$ to $(\sin (\pi r)u\mathrm{/}r,\cos (\pi r))$, and send zero to the north pole. This is continuous at zero since $\mathrm{\mid }\sin (\pi r)\mathrm{\mid }\le \pi r$, is constant at the south pole on the boundary, and is a bijection from the interior to the complement of that pole. The induced continuous bijection from the compact quotient to the Hausdorff sphere is a homeomorphism: the quotient is compact by pulling open covers back to the disk. A closed subset is compact by [F4], its image is compact by the same cover argument, and that image in the Hausdorff sphere is closed by [F4]. Identify the pole with the designated sphere basepoint. This realizes the usual two-cell based sphere.

Apply [F1] directly to $f$, and separately to the specified $H$. It gives finite subcomplexes containing their images. The finite subcomplex for $H$ automatically contains both endpoint images, since the endpoints are restrictions of $H$. This does not require selecting representatives of a family of homotopy classes, nor choosing simultaneous finite subcomplexes for such a family.

For a given based representative $f:(S^n,\ast )\to (X,x_0)$, its restriction to the source vertex is cellular because $x_0\in X^0$. Apply the finite-relative-source clause of [F2] to $(S^n,\ast )$ and $(X,\{x_0\})$. It produces a based homotopy to $g$ that is cellular. The source has dimension $n$, so $g(S^n)\subseteq X^n$. Since the homotopy fixes the basepoint, $g$ represents precisely the original based class. This applies to each class by beginning with any one representative; it asserts existence for each class and does not select representatives simultaneously.

Each homotopy just obtained, being a specified map on $S^n\times I$, also satisfies step 2.1. A basepoint outside $X^n$ cannot belong to the image of a based map landing in $X^n$, so such a skeletal conclusion would be impossible and has not been asserted. The assertions about compact images have no basepoint restriction. Zero-dimensional compact domains were treated in step 1.1; the skeletal group assertion begins at $n=1$, and no ${\pi }_0$ group law is implied. Only choice-free [F1], [F3] and the expressly choice-free clause of [F2] have been used.