Suppose I have a CW complex $X$ with skeleta $(X_n)_{n\ge 0}$ such that $\pi_k(X)=0$ for all $k\ge 0$. I want to conclude that $X$ is contractible without invoking Whitehead's theorem.
It would be enough to see that the identity $X$ is nullhomotopic. My strategy would be the following:
- Show that all inclusions $i_n:X_n\hookrightarrow X$ are null-homotopic,
- Conclude somehow that the limit $\varinjlim\, (i_n)=\mathrm{id}$ is null-homotopic.