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Math Help - Convergence of infinite sequence of reals

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    Convergence of infinite sequence of reals

    Let be an infinite sequence of reals. Show that if is convergent then we must have . If , must be convergent?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Pythagorean12 View Post
    Let be an infinite sequence of reals. Show that if is convergent then we must have . If , must be convergent?
    What have you tried? Note that \left|a_n-a_{n-1}\right|\le\left|a_n-L\right|+\left|a_{n-1}-L\right|. For your second question, what about a_n=n+\frac{1}{n}\implies a_{n}-a_{n-1}=\frac{1}{n}-\frac{1}{n-1}?
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    Quote Originally Posted by Pythagorean12 View Post
    If , must be convergent?
    You could try a_n = \sqrt n or a_n = \log n.
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