Real Analysis I · Sequences & Series
Summable increments force sequence convergence
Problem
Suppose $\sum_{n=1}^{\infty}|a_{n+1}-a_n|$ converges. Prove $(a_n)$ converges, and show that for $m>n$, $$|a_m-a_n|\le\sum_{k=n}^{m-1}|a_{k+1}-a_k|.$$
Hint
For $m>n$, write $a_m-a_n=\sum_{k=n}^{m-1}(a_{k+1}-a_k)$.
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