I was given the following exercise in a course on statistical inference:
Show that if $Y_n \overset{d}{\to} Y$ and $Y_n - Y_n' \overset{P}{\to} 0$, then $Y_n' \overset{d}{\to} Y$.
I did read a proof of this result relying on the portmanteau theorem prior to taking this class. This theorem however, has not been introduced yet in class nor in the course textbook, so we shouldn't have to rely on that result for a proof. I did try to do a more direct proof of this result, but this attempt only ended up working after I used the fact that the set of continuity points of a cdf is dense in the real numbers, another fact that has yet to be introduced to the reader.
Therefore, I'm wondering if one can prove this statement in a straightforward manner without relying on any rather involved results from probability theory. Keep in mind that at this point in the textbook, only the definitions of convergence in probability and convergence in distribution have been introduced to the reader.