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If $(a_n)$ is a monotone increasing sequence bounded above,

$(b_n)$ is a monotone decreasing sequence bounded below,

$x_n=a_n+b_n$,

Show that $\sum |x_n-x_{n+1}|$ is convergent.

My attempt: Since $(a_n)$ and $(b_n)$ are convergent,they are Cauchy sequences.So $ |x_n-x_{n+1}|\leq |a_n-a_{n+1}|+ |b_n-b_{n+1}|$ which tends to 0.However this is obviously not sufficient.Moreover the monotonic nature of $a_n,b_n$ have not been used.

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HINT: observe that $|x_{n+1}-x_n|=|a_{n+1}+b_{n+1}-a_n-b_n|\leq|a_{n+1}-a_n|+|b_{n+1}-b_n|$. Now what is $\sum_{n\geq1}|a_{n+1}-a_n|$? You should use monotonicity and boundedness here, and similarly for $\sum_{n\geq1}|b_{n+1}-b_n|$.

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