I think you mean delta = epslion/12 not 12/epsilon

Ibelieveyou are right that it is a typo--as with all epsilon-delta proofs they start "Let epsilon be given......." I am pretty sure he is talking about delta.

You are assuming r << 1 which is obviously a good assumption since we are interested in r near 0.

However to cross all the ts take delata = min (e/12,1)