# Sequences

• Mar 10th 2009, 06:04 PM
Chief65
Sequences
Need help proving....
Suppose that (an), (bn), and (cn) are sequences such that an≤ bn ≤ cn for all n ɛ N and such that lim an = lim cn = b. Prove that lim bn = b.
• Mar 10th 2009, 09:14 PM
ThePerfectHacker
Quote:

Originally Posted by Chief65
Need help proving....
Suppose that (an), (bn), and (cn) are sequences such that an≤ bn ≤ cn for all n ɛ N and such that lim an = lim cn = b. Prove that lim bn = b.

You know that $\displaystyle a_n \leq b_n \leq c_n \implies a_n - b \leq b_n - b \leq c_n - b$.
For $\displaystyle \epsilon > 0$ there are $\displaystyle N_1,N_2$ so that if $\displaystyle n\geq N_1,N_2$ we know $\displaystyle |a_n-b| < \epsilon, |c_n - b| < \epsilon$.

Let $\displaystyle N=\max (N_1,N_2)$. Try to argue that if $\displaystyle n\geq N$ then by above it must be the case that $\displaystyle |b_n - b| < \epsilon$.
• Mar 12th 2009, 07:48 AM
GaloisTheory1
Quote:

Originally Posted by Chief65
Need help proving....
Suppose that (an), (bn), and (cn) are sequences such that an≤ bn ≤ cn for all n ɛ N and such that lim an = lim cn = b. Prove that lim bn = b.

This is called the squeeze theorem for sequences. The proof is very standard.