Bastiante study this theorem
Theorem
If a sequence ( Xn ) of real numbers converges to a real number x then any subsequence of X also converges to x.
Proof:
Let ε >0 be given and let K(ε) be such that if n>= K(ε) , then | xn - x|<ε . If r1<r2<r3…<rn <… is an increasing sequence of natural numbers, it can easily be proven by induction that rn>=n . Hence , if n >=K(ε) we also have rn>=n>=K(ε) so that |xrn – x| <ε . This proves that the subsequence ( Xrn ) also converges to x , and the theorem is proved.
MINOAS