Need help proving....

Suppose that (an), (bn), and (cn) are sequences such that an≤ bn ≤ cn for all n ɛNand such that lim an = lim cn = b. Prove that lim bn = b.

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- Mar 10th 2009, 06:04 PMChief65Sequences
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 PMThePerfectHacker
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 AMGaloisTheory1