# Convergent subsequences

• Oct 20th 2009, 03:49 PM
cgiulz
Convergent subsequences
Which of these sequences b_n always has a convergent subsequence, regardless of what a_n is? Indicate reasoning.
For those subsequences b_n which do not always have a convergent subsequence, give a example; that is produce a sequence a_n such that the corresponding b_n has no convergent subsequence.

(a) b_n = cos^2(a_n) (b) bn = a_n/(1 + a_n) (c) b_n = 1/(1 + |a_n|)
• Oct 20th 2009, 04:04 PM
Jose27
Quote:

Originally Posted by cgiulz
Which of these sequences b_n always has a convergent subsequence, regardless of what a_n is? Indicate reasoning.
For those subsequences b_n which do not always have a convergent subsequence, give a example; that is produce a sequence a_n such that the corresponding b_n has no convergent subsequence.

(a) b_n = cos^2(a_n) (b) bn = a_n/(1 + a_n) (c) b_n = 1/(1 + |a_n|)

$\displaystyle \vert b_n \vert \leq 1$ for (a),(b),(c) so by Bolzano-Weierstrass they have a convergent subsequence