Sequences

**Chief65** · March 10th 2009, 06:04 PM

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.

**ThePerfectHacker** · March 10th 2009, 09:14 PM

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

Let $N=\max (N_1,N_2)$ . Try to argue that if $n\geq N$ then by above it must be the case that $|b_n - b| < \epsilon$ .

**GaloisTheory1** · March 12th 2009, 07:48 AM

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.

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