How would I prove the other way

Suppose that z_n, x_n () and z = x y. then lim_n rightarrow z_n = z doubleheadarrow lim_n rightarrow and lim_n rightarrow y_a = y.

Solution

Just reverse the steps !

We may assume that x[n] and y[n] tend to 0.

|z[n]|²= |x[n]+iy[n]| ²= x[n]² +y[n]² , and the limit of the last sum goes to zero as n tends to infinity.

So z[n] tends to 0 as n tends to infinity.