Definition P2.1. A sequence in a metric space is said to converge if there is a point with the following property: for every there is an integer such that implies that . In this case we also say that converges to , or that is the limit of , and we write , or . If does not converge, it is said to diverge.
Note. The sequence converges to in , but in , does not converge. Thus it is important to specify that “ is convergent in .”
If the range of a sequence is bounded, we say is bounded. The range of a sequence, bounded or unbounded, may be infinite or finite.
Examples
Assume all the following are in .
Example P2.2. If , then . So is bounded and has an infinite range.
Example P2.3. If , then is divergent, unbounded, and has infinite range.
Example P2.4. If , then is divergent, bounded, and has finite range.
Basic properties of convergence
Theorem P2.5. (a) converges to if and only if every neighborhood of contains for all but finitely many . ”There are infinitely many points arbitrarily close to and only finitely many points outside of any radius around iff converges to .”
(b) If , , and converges to and to , then . ”If a sequence converges to two limits, then those limits are the same limit.”
(c) If converges, then is bounded. ”If converges then its range .” (Shorthand; technically untrue.)
(d) If and if is a limit point of , then there is a sequence in such that . ”Duh.”
Proof. (a), : Suppose and let be a neighborhood of . For some , the conditions , imply . Then, corresponding to this , there exists such that implies . Thus implies .
(a), : Suppose every neighborhood of contains all but finitely many of the . Fix , and let be the set of all such that . By assumption, there exists (corresponding to this ) such that if . Thus if ; hence .
(b) Suppose and , and let be given. There exist such that
Hence for we have
and since is arbitrary, .
(c) Let . There exists such that . Put
Then for .
(d) For each positive integer , there is a point such that . Given , choose so that . If , it follows that . Hence . ∎
Note. Return to Rudin p. 49.