@Shadock The second one means $\lim_{x\to y}|f(x)-f(y)|\le0$, which is true iff $f$ is continuous at $y$. I feel like this is not what you meant to say.
I think you meant to say that, $|f(x)-f(y)|\le|x-y|$ is true for all $x$ in some neighborhood of $y$
which neither of the two things you wrote mean
(The first one doesn't really mean anything, actually; I'm pretty sure the symbols can't be put together like that)
('cause on the left-hand side, $x$ is a dummy variable, and on the right, it's not)
@BalarkaSen Beginning of real analysis, but I haven't really started yet