So for a sequence of functions $\{ f_n\}_{n \in \mathbb{N}}$ then
$f_n$ converges pointwise to some function $f$ if $$\forall x \in I, \forall \epsilon > 0, \exists N > 0 :n\geq N \implies |f_n(x) - f(x)| < \epsilon.$$
Is an equivalent condition for this:
$f_n\to f$ pointwise on $(a,b)$ if for each fixed $x\in(a,b)$, $|f_n(x)-f(x)|\to 0$ as $n\to\infty$
and for uniform convergence,
$$\forall\epsilon > 0, \exists N>0 : n\geq N \implies |f_n(x) -f(x)| < \epsilon \quad \forall x \in I.$$
Is this equivalent to:
$f_n\to f$ uniformly on $(a,b)$ if $\sup_{a< x<b}|f_n(x)-f(x)|\to 0$ as $n\to\infty$ ?
I've searched around and every PDF file I see defines the definitions with the former for each.