That's wrong. You don't know that $\displaystyle \forall m,n \in \mathbb{N}, \|x_m - x_n\|_X \leq M \|U(x_m-x_n)\|_Y$. You can get that $\displaystyle \forall m, n \in \mathbb{N}, \exists x_{m,n} \in X$ such that $\displaystyle U(x_{m,n}) = y_m - y_n$ and $\displaystyle \|x_{m,n}\|_X \leq M \|y_m - y_n\|_Y$, but that's all.

$\displaystyle x_{m,n}$ and $\displaystyle x_m - x_n$ can differ by anything in $\displaystyle \text{Ker}(U)$, which is why I used the quotient norm, but there has to be a more direct method.