- #1

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__Attempt at answer__

__<=__

Assume d(x,x') < r'-r.

Assume a is an element of B(x,r).

Then d(x,x') < r'-r implies d(x,x') + r < r' implies d(x,x') + d(x,a) < r' as d(x,a) < r.

This implies that d(x',a) < r' by use of the triangle inequality, and thus a is an element of B(x',r').

__=>__

Let a be an element of B(x,r).

Then a is an element of B(x',r').

So d(x,a)<r and d(x',a)<r'.

This implies that d(x',a)-d(x,a) < r'-r

This would almost suggest a use of the reverse triangle inequality, but the inequality is pointing the wrong direction to use here. I am not sure how to proceed.