Let

be an enumeration of the set

. Show that there exists a subsequence

such that

.

**Proof**
Choose

such that

. (This is possible by the use of the Archemedian property.) Choose

such that

is the smallest natural number that satisfies

. Then

and

. Now choose

such that

is the smallest natural number that satisfies

. Then

and

. Assume

have been selected such that

and

. Select

such that

is the smallest natural number that satisfies

. This means

and

. By induction,

and

for all

. So given

, choose

. And so

implies

. Hence,

.

**Q.E.D.**
Is this proof correct; can I construct the subsequence in such a way? And if not, how should I go about this proof? Thanks.