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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
numbers. Then (snk )∞ is called a subsequence of (sn ).
For convergent sequences, the subsequences must have the same asymptotic behavior as the entire
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
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 ). ...
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