I have to be honest that I am very lost on the same kind of problem about proving existence of smooth function. I have not done much topology, hence unfamiliar with the "covering" business.
In the proof of problem 4 of the provided solution. The solution is brief and I am having trouble in writing the mathematics in detail. Could anyone help in providing a more complete/detailed solution. I am very unclear (in term of writing the mathematics formally) in the sentence "For each $i$, we apply problem 2 and find a $w_i\equiv 1$ on $W_i$ and $0$ near the boundary $\partial V_i$." I know this is the $C^{\infty}$ version of Usysohn's lemma. But I am having difficulty in linking things well in writing proper sentences due to my lack of training in topology.
enter link description here Appreciate for any helps.