Continuity proofs and Lipschitz-continuity

Question

I found quite some proofs of the continuity of a function $f \in Y^X$, with $X, Y$ metric spaces, that basically proceed by showing that

$$ d(f(x), f(y)) \leq d(x,y),$$

for all $x, y \in X$, without any mention to $\varepsilon$, and $\delta$.

I have the two following questions:

1. Is implicit in these kind of proofs that we basically set $\delta := \varepsilon$ and then, by th fact that $d(x,y) < \delta$, a fortiori $d(f(x),f(y)) < \varepsilon$?
2. Is this equivalent to say that the function we have proved to be continuous is actually Lipschitz-continuous with $K=1$?

Any feedback is welcome.
Thank you for your time.