I'm very confused on the idea of sequence of functions, I feel like it's very trivial and I'm overcomplicating it.
For part a, I was thinking of constructing a family of functions ${f_n}$ such that each n would have a different limit and then the supremum would just be each of those limits. The goal is to create jump discontinuities for each n.
For part b, by defn equicontinuous means $\forall$ $\epsilon$ > 0 $\exists$ $\delta$ >0 s.t. when $|x-y| < \delta$ it follows that $\forall$ n $\in$ natural numbers $|f_n(x) - f_n(y)| < \epsilon$. Intuitively, $sup f_n(x)$ would be part of the equicontinuous family since the $sup$ cannot be more than $\epsilon$ away otherwise the defn wouldn't hold. Is this thinking correct?
Also I saw there are similar posts for part b, but I haven't done metric spaces yet.
What do you mean each $n$ has a different limit?
Nonetheless, consider the $n$ root of $|x-\frac{1}{2}|$ for part $a$. The sup will be one everywhere except at $\frac{1}{2}$ where it is zero.
The problem with the sup is that for different $n$ the delta required can shrink with the sup having no available delta.