Pointwise and uniform as i understand

Question

Verify if this statement is true: The pointwise convergence depends on each point $x$, that is you need to count and fix every  $x$ before passing to the limit. On the other hand, uniform convergence is independent of $x$. Hence, the pointwise convergence is enough to cover a countable set $Q$ in the following sence $$ \forall x \in Q, ~~~~ f_n(x) \longrightarrow f(x)$$ where we took every single $x$ into account when we passed to the limit as we covered the set by writing $\forall x$ for each $x \in Q$ and not enough to cover an uncoutable set like $[0,1]$, in such case you need a uniform convergence to generalise your convergence on $[0,1]$.

Answer 1

A simple way to think about this is as follows.

Pointwise at $x$. Given any small quantity $\epsilon$, there is a positive integer $N$ such that $f_n(x)$ differs from $f(x)$ by less than $\epsilon$ for all $n>N$.

Uniform. The same integer $N$ can be used for all $x$.

It is important to realise that convergence can occur at every $x$ but for there to be no $N$ which works for all $x$. Can you think of or construct an example?