Let $X=]0,\infty[: \{x \mapsto \arctan{(nx)}: n \in \mathbb N \}$
so $f_{n}: ]0,\infty[ \to \mathbb R, x \mapsto \arctan{(nx)}$ for all $n \in \mathbb N$
There is a statement, saying (w.r.t. $d_{\infty}$):
$1.$ $(f_{n})_{n}$ is equicontinuous
$2.$ $(f_{n})_{n}$ is not uniformly equicontinuous
$3.$ each function $f_{n}$ is uniformly continuous.
The difference between $1.$ and $2.$ is clear but I do not understand why $(f_{n})_{n}$ is equicontinuous because with every increasing $n$ my slope close to $0$ increases, so my chosen $\delta$ will need to get smaller and smaller.
And surely if $1.$ and $3.$ are true our sequence $(f_{n})_{n}$ necessarily has to be uniformly equicontinuous. Why is this not the case, any explanations are greatly appreciated.