I am working on Exercise 4.1.11 in Hatcher's Algebraic Topology text, which is as follows: Show that a CW complex $X$ is contractible if it is the union of an increasing sequence of subcomplexes $X_1 \subset X_2 \subset \cdots$ such that each inclusion $X_i \hookrightarrow X_{i+1}$ is nullhomoto...