Convergence of two Sequences of random numbers.

Question

In the prior question I proved if Z_n ⇒ Z_∞ and Y_n ⇒ C
Z_n + Y_n ⇒ Z_∞ + C
Using almost sure convergence.

But the next question asks to show 'if Y_n ⇒ Y_∞, then it's not always true that Z_n + Y_n ⇒ Z_∞ +Y_∞'

C is just a constant though, whereas Y_∞ is a random variable, not that it should matter, is the question asking me to give a specific example or prove with something I've missed? I've looked everywhere but can't seem to find even a hint.

Only seeking a push in the right direction, thanks!