Consider a sequence of functions $f_{n}:K=[0,1]\times [0,1]\to \mathbb{R},\;f_{n}(x,y)=\dfrac{\ln(e^{nx}+e^{ny})}{n}$. Is this sequence Cauchy (in $(C(K),\;\rho)$, where $\rho(f,g) = \sup\limits_{x\in K}|f(x)-g(x)|$)?
I tried to estimate function by using its convexity, however everything I tried was a little bit tedious. Is there a fast way to prove (or maybe disprove) that this sequence is Cauchy?
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
$$ 0\leq f_n(x,y)-f(x,y) \leq \frac{\ln 2}{n} .$$