If z_n = r_n e^{ix_n} converges to z = re^{ix}, one may ask why is it that x_n converges to x? One can show this directly using the fact that the (z_n) sequence converges iff its real and imaginary components converge. It is fairly obvious that r_n converges to r (use ||x|-|y||<=|x-y|). Using e^{ix} = cosx + isinx one can get an inequality involving sin(x_n), cos(x_n), cosx, and sinx, then an inequality in x_n,x alone that shows x_n converges to x.

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