Am I right with my arguments or is something wrong?
proof. Let $(x_n)_{n \in \mathbb{N}}$ a Cauchy sequence, then for every $\epsilon > 0$ exists an $N \in \mathbb{R}$ such that for every $n,m \geq N$, we have $$d_M (x_n, x_m) < \epsilon$$ Since $f$ is continuous, given $\epsilon > 0$ we can take $\delta = \epsilon$, then $$d_M(x_n, x_m) < \delta = \epsilon$$ So $$d_N (f(x_n),f(x_m)) < \epsilon$$ Therefore, $(f(x_n))_{n\in\mathbb{R}}$ is a Cauchy sequence.
There's a counterexample to the claim in the comments already. To answer your question about where you might have gone wrong, you should examine why you think you're allowed to choose $\delta = \epsilon$ in your argument.
The definition of $f$ being continuous at $x_n$ tells you that for each $\epsilon > 0$ there is a $\delta > 0$ such that for all $y$, if $d(x_n, y) < \delta$ then $d(f(x_n), f(x_m)) < \epsilon$. It does not say that you get to pick the $\delta$. As mentioned in the comments, it also does not say that the same $\delta$ will work for all $n$.