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**FLT** I want to prove that for any ball $\displaystyle B_r(a)$, one has $\displaystyle \text{diam} B_r(a) \leq 2r$

So far I know the following:

$\displaystyle \text{diam} B_r(a) = \sup B_r(a)$. My problem is, what on earth does $\displaystyle \sup B_r(a) = \sup\{\rho(x,a) < r | x,a \in X\}$ actually mean; how do I go about proving that the supremum is indeed less than or equal to 2r?

The statement seems obvious, but the proof for me is a little tricky because of this part. Thanks.