Prove that {$a_n$} converges 
Let {$a_n$} be a sequence of real numbers. Suppose that for every pair of integer $N>M>0$, it holds that $\lvert a_M - a_{M+1} \rvert$+$\lvert a_{M+1} - a_{M+2} \rvert$+$...$+$\lvert a_{N-1} - a_{N} \rvert$$\le$1
Prove that {$a_n$} converges.

I tried to use properties of Cauchy sequence (but I think it is not guaranteed that it is Cauchy sequence) and the fact that sum of absolute values are bounded by 1. I want to know how to approch a problem..
 A: Consider the sequence of differences $b_n \stackrel{\rm def}{=} a_{n+1}-a_n$. Then, we have, for any $N\geq 1$
$$
a_{N+1} - a_1  = \sum_{n=1}^N b_n
$$
so it is sufficient to show the series $\sum_{n} b_n$ converges. But your assumption actually implies it converges absolutely, which is even stronger. Indeed, we have
$$
\sum_{n=1}^N \lvert b_n \rvert \leq 1
$$
for all $N\geq 1$, so that $(\sum_{n=1}^N \lvert b_n \rvert)_N$ is a non-decreasing sequence* of real numbers bounded above. The monotone convergence theorem yields the conclusion.
(*) Since $\sum_{n=1}^{N+1} \lvert b_n \rvert - \sum_{n=1}^N \lvert b_n \rvert = \lvert b_{N+1} \rvert \geq 0$.
A: This is not a very neat proof and may be there is much more simple but it works.
Let's suppose that $(a_n)$ is not a Cauchy sequence, then :
$$\exists \epsilon > 0, \forall n \geq m, \exists p>q>n, |a_q-a_p| \geq \epsilon$$
For $m=0, \exists p^0>q^0>0$ such that :
$$\epsilon \leq |a_{q^0}-a_{p^0}|=|a_{q^0}-a_{q^0+1}+a_{q^0+1}...-a_{p^0}| \leq |a_{q^0}-a_{q^0+1}|+|a_{q^0+1}-a_{q^0+2}|+...+|a_{p^0-1}-a_{p^0}|$$
Now let's take $m=p$, by the definition $\exists p^1>q^1>m$
$$\epsilon \leq |a_{q^1}-a_{p^1}| \leq |a_{q^1}-a_{q^1+1}|+|a_{q^1+1}-a_{q^1+2}|+...+|a_{p^1-1}-a_{p^1}|$$
By induction, we construct two sequences $(p^{(n)})$ and $(q^{(n)})$, and for every $n$, we have :
$$\epsilon \leq |a_{q^{(n)}}-a_{p^{(n)}}| \leq |a_{q^{(n)}}-a_{q^{(n)}+1}|+|a_{q^{(n)}+1}-a_{q^{(n)}+2}|+...+|a_{p^{(n)}-1}-a_{p^{(n)}}|$$
and
$$p^{(n)} > q^{(n)} > p^{(n-1)}>q^{(n-1)}$$
Let's choose $n > 1/\epsilon$, we have :
$$ 1<n \epsilon \leq |a_{q^0}-a_{p^0}|+|a_{q^1}-a_{p^1}|+...+|a_{q^{(n)}}-a_{p^{(n)}}| \leq |a_{q^0}-a_{q^0+1}|+...+|a_{p^{(n)}-1}+a_{p^{(n)}}|$$
We have a contradiction, then $(a_n)$ is a real Cauchy sequence, then converges.
