lec14 - Math 117 14 Subsequences Subsequences Denition 14.1 Let(sn be a sequence and let(nk be a strictly increasing sequence of natural n=1 k=1 numbers

# lec14 - Math 117 14 Subsequences Subsequences Denition 14.1...

This preview shows page 1 out of 1 page.

Math 117 – May 19, 2011 14 Subsequences Subsequences Definition 14.1. Let ( s n ) n =1 be a sequence, and let ( n k ) k =1 be a strictly increasing sequence of natural numbers. Then ( s n k ) k =1 is called a subsequence of ( s n ). For convergent sequences, the subsequences must have the same asymptotic behavior as the entire sequence. Theorem 14.2. If ( s n ) converges to s , then every subsequence of ( s n ) converges to the same limit s . However, even if a sequence does not converge, it may have convergent subsequences. Theorem 14.3. Every bounded sequence has a convergent subsequences. On the other hand, if a sequence is not bounded, it must have subsequences diverging to positive or negative infinities. Theorem 14.4. Let ( s n ) be a sequence of real numbers. ( a ) If ( s n ) is unbounded above, then it has an increasing subsequence diverging to + . ( b ) If ( s n ) is unbounded below, then it has a decreasing subsequence diverging to

#### You've reached the end of your free preview.

Want to read the whole page?

• Fall '08
• Akhmedov,A
• Math, Limit of a sequence, subsequence, lim sup sn, lim inf sn
Stuck? We have tutors online 24/7 who can help you get unstuck.
Ask Expert Tutors You can ask You can ask You can ask (will expire )
Answers in as fast as 15 minutes