Prove that for continuous f, f(x_n)>=f(y_n) => f(lim x_n)>=f(lim y_n)

I started a proof by contradiction, because that is how I remember doing these types of problems in the past. I said, suppose not, then there is an N such that for n>N, f(y_n)>f(x), but how can I go from there to showing that means there is an n such that f(x_n)>f(y_n). I have a feeling that it is a clever choice of epsilon that I cannot think of.

Thanks!