lec14 - Math 117 – May 19, 2011 14 Subsequences...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 117 – May 19, 2011 14 Subsequences Subsequences Definition 14.1. Let (sn )∞ be a sequence, and let (nk )∞ be a strictly increasing sequence of natural n=1 k=1 numbers. Then (snk )∞ is called a subsequence of (sn ). k=1 For convergent sequences, the subsequences must have the same asymptotic behavior as the entire sequence. Theorem 14.2. If (sn ) converges to s, then every subsequence of (sn ) 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 (sn ) be a sequence of real numbers. (a) If (sn ) is unbounded above, then it has an increasing subsequence diverging to +∞. (b) If (sn ) is unbounded below, then it has a decreasing subsequence diverging to −∞. We next consider all the limits that subsequences of a given sequence can converge to. Definition 14.5. Let (sn ) be a bounded sequence. A partial limit (or subsequential limit) of (sn ) is any real number that is the limit of some subsequence of (sn ). Let S be the set of all partial limits of (sn ), then the limit superior (or upper limit) of (sn ) is defined to be lim sup sn = sup S. Similarly, the limit inferior (or lower limit) of S is defined to be lim inf sn = inf S. It is an interesting question whether the lim sup sn and lim inf sn must themselves be partial limits of (sn ), which is answered by the following. Theorem 14.6. Let (sn ) be a bounded sequence and M = lim sup sn , m = lim inf sn . Then M and m are themselves partial limits of (sn ). ...
View Full Document

Ask a homework question - tutors are online