1) Can you find a sequnce of real nos. {sn} which has no convergent subsequence and yet for which {|sn|} converges.

I am not able to think of such a sequence.:(

2) If a subsequence of {sn} of terms with even subscripts converges to L, as well as the subsequence with odd subscripts (converges to L i.e.) then, prove that the given sequence converges to L.

Here I intuitively know the theorem statement is sort of obvious, but how do I go about the proof?

Any help will be appreciated.