I'm tearing my hair out trying to prove that a contractible subcomplex ,$K$, of a contractible CW complex, $L$, is a strong deformation retract of $L$.
What I have so far:
I can show that the contractibility of K gives a retract $ L \rightarrow K$.
I also think I can show that K is a (not necessarily strong) deformation retract of L.
2.1 I also know it to be a fact that a subcomplex is a def retract iff it's also a strong def retract but I dont know how to prove this.
3.Lastly I've also reduced the problem to the case where $K$ is a point as follows: since $K$ is a subcomplex, a def retract of $K$ onto $p\in K$ extends to a homotopy of $L$. Then if the statement held for K a point, we could perform a subsequent strong def retract onto $p$, furnishing a contraction of $L$ which maps $K$ to itself at each stage of the homotopy. This provides a contraction of the space $K\times I \cup L\times 0 \cup L\times 1$. But this space is a subcomplex of the complex $L\times I$ and so $L\times I$ retracts onto$K\times I \cup L\times 0 \cup L\times 1$. The statement follows.
So yeah I'd be very grateful for any hints or solutions. Thanks for your time.