Sequences, series, summability Proposition

Subsequences of a convergent sequence converge to the same limit

Builds on Subsequence Limit of a sequence Epsilon-N definition of the limit of a sequence

Statement

Let $(a_n)$ be a real sequence that converges to a real number $L$ . Let $(a_{n_k})_{k \in \mathbb{N}}$ be a subsequence of $(a_n)$ . Then $a_{n_k} \to L$ as $k \to \infty$ .

Remarks

For the proof, let $\varepsilon > 0$ and take $N$ with $|a_n - L| < \varepsilon$ for $n \ge N$ . Since $n_k \ge k$ , we get $|a_{n_k} - L| < \varepsilon$ for all $k \ge N$ . It gives a test for divergence. If two subsequences have different limits, the sequence diverges. For $(-1)^n$ the even terms tend to $1$ and the odd terms tend to $-1$ .

Reading path

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Sequence definition
Real sequence definition
Subsequence definition
Limit of a sequence definition
Epsilon-N definition of the limit of a sequence statement
Subsequences of a convergent sequence converge to the same limit statement

Tag	`0VHAVXW08V`
URI	`m2://0VHAVXW08V`
Kind	statement, proposition
Writing	W0 stub
MSC	40A05
Status	This copy of the site has no build data. Run `m2 status 0VHAVXW08V`.
Uses	Subsequence, Limit of a sequence, Epsilon-N definition of the limit of a sequence

Subsequences of a convergent sequence converge to the same limit

Statement

Remarks

Reading path

Further reading

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