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Math Help - convergent sequences proof

  1. #1
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    convergent sequences proof

    Prove the following by using this definition:
    A sequence (Sn) is said to converge to the real number S provided that for each e > 0 there exists a real number N such that for all n element of N, n > N implies that |Sn - S| < e
    If (Sn) converges to S, then S is called the limit of sequence (Sn), and we write lim n -> infinity Sn = S, lim Sn = S, or Sn -> S. If a sequence does not converge to a real number, it is said to diverge

    Prove this from the definition above:
    a) lim (3n + 1)/(n + 2) = 3
    b) lim (sin n) / n = 0
    c) lim (n+2)/(n^2 - 3) = 0
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  2. #2
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    Quote Originally Posted by luckyc1423 View Post
    a) lim (3n + 1)/(n + 2) = 3
    For any e>0 choose N=5/e

    Then if n>N we have n>5/e thus, 5/n<e.

    But then,

    |(3n+1)/(n+2)-3|=|-5/(n+2)|=5/(n+2)<=5/n<e
    b) lim (sin n) / n = 0
    For any e>0 choose N=1/e then if n>N we have 1/n<e.

    Thus,
    |sin(n)/n-0|=|sin(n)/n|<=1/n<e
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  3. #3
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    Quote Originally Posted by luckyc1423 View Post
    c) lim (n+2)/(n^2 - 3) = 0
    For any e>0 choose N=max{2,4/n}

    Then if n>N* we have 4/n<e.

    But then!

    |(n+2)/(n^2-3)|=(n+2)/(n^2-3)<=2n/(n^2-3)<=2n/(.5n^2)=4/n<e

    Q.E.D.

    *)You also need to consider the case max{2,4/n}=2.
    But that case is easy.
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