Preserving convergences under composition of function sequences

I have following propositions to decide whether they are true or false

(Also checked some problems in site such as Question1 ,Question2)

Let $g:A\subset \mathbb{R} \to \mathbb{R}$ bijection and for all $n\in \mathbb{N}$, $f,f_n : g(A)\to \mathbb{R}$ be functions.

1. If $f_n \to f$ then $f_n \circ g \to f\circ g$

2. If $f_n \rightrightarrows f$ then $f_n \circ g \rightrightarrows f\circ g$ (here $\rightrightarrows$ means uniformly convergence)

For the first one I tried following:

Let $x \in A$ and $\varepsilon >0$ is given. Because $f_n \to f$, there exists a $n_0\in \mathbb{N}$ such that for all $n\geq n_0$ we have $|f_n(x)-f(x)|<\varepsilon$. Here if we choose $N=n_0$ then for all $n\geq N=n_0$ we have

($\exists ! y \in g(A) : g(x)=y$)

$| (f_n \circ g)(x) - (f\circ g)(x)|=| f_n( g(x))- f(g(x)|= |f_n(y)-f(y)|<\varepsilon$.

I think the second one is wrong. Tried they way in the link Question2, but $g$ may be unbounded. Therefore can't find a counterexample.

The further question may be which properties that $(f_n)$ has is preserving under bijection such that boundedness, equicontinuity, uniform continuity, etc.,

Thanks in advance for your help and guidance.