Is this sequence Cauchy?
Solution 1:
After you revised the question, I think that the sequence is Cauchy. Since $C(K)$ with that metric is complete, you can conversely show that $f_n$ converges uniformly to a function after finding it. Going by that reasoning, you first need to find the limiting function. Note that
$$ \max\{ x,y \} \leq f_n(x,y)\leq \max\{ x,y \} +\frac{\ln 2}{n}. $$
So the limiting function should be $f(x,y)=\max\{ x,y \}$. Look at the difference $ f_n -f$, and you see that
$$ 0\leq f_n(x,y)-f(x,y) \leq \frac{\ln 2}{n} .$$