Question

Let {an}∞n=1 be a sequence converges to a limit L ∈ R. Prove that any subsequence of {an}∞n=1 is convergent and converges to the same limit L.

Answer 1

<a_n> be a sequence converges to L then by definition of convergence,

$\in$ > 0 is given, $\exist$ n$\in$ N such that |a_n - L| < $\in$ for all n $\geq$ N

let <b_n> be any subsequence of <a_n>

$\because$ b_n = a_m for some m $\geq$ n $\geq$ N

consider,

|b_n - L| = |a_m - L| < $\in$ for all n $\geq$ N

|b_n - L| < $\in$ for all n $\geq$ N

$\implies$ by definition of convergence <b_n> is also convergent.

also <b_n> converges to L.