** There is a subs.

converging to

. Choose now

.

** There is a subs.

converging to

. Choose now

in such a way that

(we can do this because we've an infinite subsequence here and thus there exists

!)

** There is a subs.

converging to

. Choose now

in such a way that

(we can do this because we've an infinite subsequence here and thus there exists

!)

Continue inductively in the above way, and in the m-th step:

** There is a subs.

converging to

. Choose now

in such a way that

(we can do this because we've an infinite subsequence here and thus there exists

!)

Take it from here...

Tonio