Let $A$ be a non-empty compact subset of $X$. Prove that there exist points $a,b \in A$ such that $d(a,b) = \sup \left\{ d(x,y):x,y \in A \right\}\,.$
The question was split into two pieces with us needing to show:
| d(x,y) - d(x',y') | \leq d(x,x') + d(y,y')
My attempt was to say that if $A$ is compact $\exists a,b \in A$ such that there are Cauchy sequences $ \left\{ x_n \right\} , \left\{ y_n \right\} $ such that for $ \epsilon_1, \epsilon_2 > 0,\, d(x_n,a) < \epsilon_1\text{ and }d(y_n,b) < \epsilon_2 \,$.
Using this I can say:
$ | d(x_n,y_n) - d(a,b) | \leq d(x_n,a) + d(y_n,b) < \epsilon_1 + \epsilon_2 = \epsilon_3$
$\square$
Is this completely wrong?