Both these theorems were discussed in class, I wanted to discuss some doubts regarding theorem 1. Is it possible to say simply from theorem 1 alone that any sequence for which $\left|x_{n+1}-x_n\right|\rightarrow 0$ is a cauchy sequence? I can most definitely say $\alpha ^n\rightarrow 0$ as $0<\alpha <1$ so that would seem to imply that if a sequence satisfied the condition $\left|x_{n+1}-x_n\right|\rightarrow 0$ it would be cauchy. But at the same time the sequence $\sqrt{n}$ satisfies this condition but is obviously not cauchy as it is not convergent. Could anyone explain what exactly is causing this mismatch and maybe I am assuming more than I was given.