${Let \ x _n \ be \ a \ sequence \ of \ integers \ ; \ x_n \in Z \ , \ \forall n \in N }$ ${ . If \ x_n \ is \ cauchy \ , \ prove \ that \ x_n \ is \ eventually \ constant \ }$
${(i.e , \exists c\in Z , \exists k \in N \ s.t \ x_n =c , \forall n \geq k )}$
2. Take $\epsilon =\frac 12$ in the definition of a Cauchy sequence.
3. Assume that the $x_n$ don't eventually become constant. That is, for all integers k, $|x_i - x_j| > 0$ for some integer i and some integer j greater than k.
Since the sequence is Cauchy, there is an integer k such that $|x_n - x_m| < \frac{1}{2}$ for all n and m greater than k. From our assumption, we can take n = i and m = j. Then $0 < |x_i - x_j| < \frac{1}{2}$. This obviously can't be true, as the difference between two integers must be an integer itself. QED