Subsequence
A subsequence of is where is a strictly increasing sequence of natural numbers.
Definition 1.14 (Subsequence)
A subsequence of a sequence is a sequence of the form where is a strictly increasing sequence of natural numbers.
Limits along subsequences
Lemma 1.15
If , then any subsequence must converge to the same limit.
Proof
Since implies , induction gives for all .
Given , take with for all . Then forces , so . Hence : every subsequence of a convergent sequence converges to the same limit.
Consequently, a sequence whose odd and even subsequences both converge to the same limit itself converges to : for , whichever parity has, the corresponding estimate applies.
Related
Stated in
- Definition 1.14 (Subsequence)§1.2 Bolzano-Weierstrass Theorem
- Lemma 1.15§1.2 Bolzano-Weierstrass Theorem