Prove that
if
Proof. Since
we know that for all there exists a positive integer
such that
for all . In particular, if
then there exists an integer
such that
So, for any take
then we have
for all . (Since
implies
and
.) Then, since
implies
we have
for all . Hence,
Maybe instead of for any $\epsilon > 0$, you could have for any $\sqrt{\epsilon} > 0$.