The solution to number 5 of this released exam seems rather sophisticated to me. I would have said:
The dimension of $S^1 \times R$ is $3$ as the $\dim(S^1 \times R) = \dim(S^1) + \dim(R) = 2 + 1 = 3$ but $\dim(R^2) = 2$, as they do not have the same dimension, there cannot exist a bijection between the two spaces thus there can be no diffeomorphism between the two.
I felt relativively confident about my answer originally, but the answer seems so long that I have doubts that I am correct. Of course, she says there are multiple ways to show this, but the way she presents is rather involved and makes me concerned that I have ommitted necessary rigour (as I have done before).
Furthermore, if I did do it incorrectly, is there a way to correct my reasoning to make it rigourous?