General fact:  Let $A_{-1} \subset ... A_n \subset ..$ be a filtration of cellular inclusions of $CW$ complexes.(More generally let this be a filtration of cofibrations).  Then $A_n$ contractible in $A_{n+1}$ $\implies$  $A:=colim_n A_n$ is contractible. (Here $A$ is given the weak topology)

> Proof:  Consider the composition $A_n \times I \to A_{n+1} \to A$. 
> Since this is a cofibration, extend it to a map $A \times I
 \xrightarrow{\alpha_n} A$.  The map $f: A\times I \to A$ defined by
> $f|_t=\alpha_{n+1}(2^{n+1}
 t-2^{n+1}+2)\circ\alpha_n(1)...\circ\alpha_1(1)$ for
> $1-\frac{1}{2^n}\leq t \leq 1-\frac{1}{2^{n+1}}$, is the required
> retraction.  It is continuous because $f|_t$ is continuous when restricted to each $A_n$ and obviously $f|_a$ is continuous for all $a \in A$.

$S^\infty$ satisfies the hypotheses of this general fact:  $S^n \to S^{n+1}$ is contractible  - $S^{n+1}$ can be given an $n-skeleton$ that is a point, and cellular approximation implies that this map is this inclusion is homotopic to the null map.