Uniform convergence on interval, difference between $\forall N: x, y \in [0, N[$ vs just $x,y \in [0, \infty[$

Question

I am wondering whether the two following statements are equal (are they both uniform convergence for the same interval?):

$$ \forall n \forall \epsilon \exists \delta \forall x,y \in [0, n[: |x-y| < \delta \implies |f(x) - f(y)| < \epsilon $$

and uniform convergence (for an interval):

$$ \forall \epsilon \exists \delta \forall x,y \in [0, \infty[: |x-y| < \delta \implies |f(x) - f(y)| < \epsilon $$

What are the differences?

Answer 1

The difference is that in the first case you essentially have a family of statements (one for each interval $[0,n]$), while in the second case you have a single statement (about the single interval $[0,\infty)$).

It's like saying a continuous function $f:[0,\infty)\rightarrow[0,\infty) $ is bounded on every finite interval $[0,n]$ (which is true) versus saying $f$ is bounded on $[0,\infty)$ (not necessarily true).