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Let $\{X_n\}$ be a sequence of real numbers and let $Y_n= X_{n+1}-X_n$, then $\{Y_n\}$ converges if and only if the sequence $\{X_n\}$ converges.

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    $\begingroup$ Difficult to prove "only if" $\endgroup$ – Henry May 3 '16 at 18:01
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    $\begingroup$ $\sum_1^NY_n=X_{N+1}-X_0$. Now think what it means for $\sum Y_n$ to converge and for $X_n$ to converge. $\endgroup$ – almagest May 3 '16 at 18:01
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    $\begingroup$ Did you want to write "the series $\sum_{n=1}^\infty Y_n$ converges" instead of "$\{Y_n\}$ converges? $\endgroup$ – Martin Sleziak May 4 '16 at 4:13
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Consider the sequence $X_n = n$. Then $\{X_n\}$ does not converge, but $\{Y_n\}$ does as $Y_n = 1\forall n$, this is a contradiction to what you want to prove.

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