I found a very elegant construction in this link: https://math.stackexchange.com/a/311289/444015, I have no questions about this example. However, at least for me, it doesnt look like a natural construction.
I tried to find more direct examples, but I didnt succeed. Every smooth equicontinuous sequence that I found is bounded.
I was wondering if there is any more obvious example than this, or if indeed to find any example, it is necessary to construct the sequence of functions in a similar way.