A text I'm using passes off the following statement as obvious, but I can't see where their logic came from,
As $X_n \le x - c - \epsilon$ and $Y_n \le c + \epsilon$ imply $X_n + Y_n \le x$, it is apparent that $P(X_n \le x-c-\epsilon) - P(Y_n > c+\epsilon) \le P(X_n + Y_n \le x) \le P(X_n \le x-c+\epsilon) + P(Y_n < c-\epsilon)$
Where does this statement come from? It seems entirely out of the blue to me.