Why is this CW complex connected?
Solution 1:
If you know that $\pi_0(X,x_0)=0$, then the result is trivial. Explicitly, you can consider two points $x,y$. There is an $i$ such that $x,y\in X_i$. Since the inclusion $X_i\to X_{i+1}$ is nullhomotopic, there is an arc connecting these two points in $X_{i+1}$.