This is similar to another question asked on here, but the answers there didn't help me (and it is a slightly different set-up).
Consider the vector bundle $$E := ([0,1]\times \Bbb{R})/\sim \to S^1$$ where $\sim$ is the equivalence relation that says $(0,t) \sim (1,-t)$ for all $t \in \Bbb{R}$. Does there exist a nowhere vanishing section of $E$?
I know the answer is "no", and I have tried to follow the answers given on this post, but I don't understand how they conclude the section $s$ must have a zero, just because $F$ does (notation from the linked post).
Could someone explain this in full? Because I really lack understanding of differential geometry to be able to 'fill in gaps' in proofs like this if I'm not familiar with them.
Thanks in advance!