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Math Help - Sequences

  1. #1
    Newbie Chief65's Avatar
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    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.
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    Quote Originally Posted by Chief65 View Post
    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.
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  3. #3
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    Quote Originally Posted by Chief65 View Post
    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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