It goes as follows:
If $|x_{n+1} - x_n| < \epsilon$, $\{x_n\}$ is Cauchy. We have: $$\forall \epsilon_0 > 0, \exists N : n > N \implies |x_{n+1} - x_n| < \epsilon_0$$ Let $m,n \in \mathbb{N}$, and $m > n$. Consider: $$|x_{m} - x_n| = | x_m - x_{m-1} + x_{m-1} - x_{m-2} + ... + x_{n+1} - x_n |$$ By the triangle inequality, $$| x_m - x_{m-1} + x_{m-1} - x_{m-2} + ... + x_{n+1} - x_n | \leq |x_m - x_{m-1}| +...+ |x_{n+1} - x_n|$$ Then, for $n > N$, as $m>n>N$, $$|x_m - x_{m-1}| +...+ |x_{n+1} - x_n| < (m-n+1) \epsilon_0$$ Hence, let $\epsilon_1 = (m-n+1) \epsilon_0$. Finally, $$\forall \epsilon_1 > 0, \exists N: m,n > N \implies |x_m - x_n| < \epsilon_1$$ and the sequence is Cauchy, as intended.
This is untrue, the counterexample being the harmonic series. But each step of the way seems to make sense, I can't seem to figure out why it is wrong.
Your assertion about $\epsilon_1 = (m-n+1) \epsilon_0$ depends on $m$ and $n$ ! That is your mistake.