Prove or Disprove If xn converges then for any subsequence x

Prove or Disprove:

If x_n converges then, for any subsequence x_nk, the subsequence converges and the limits are the same.

Solution

Let a sequence xn converge to L.

Then |xn-L|< epsilong for n >=M

This is true for all terms in xn

As xnk a term is in subsequence it also lies in xn

Hence satisfies

|xnk-L|

From this we conclude that the subsequence converges.

But please note that limits need not be the same.

The subsequence may have a limit which is less than L.

THe only point we can say is subsequence converges to a limit <= L.