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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?

  • 1
    Where do you get $a, b$ from?2011-12-09

1 Answers 1