I know both definitions but I was wondering what are the relations between them. My question is if someone could explain intuitively the differences between these types of convergence. Specifically, is it true that convergence in the uniform norm means uniform convergence.
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Explanation of 1: being able to find $N$ such that $|f_n(x) - f(x)| \le \epsilon $ for all $x$ (in some interval) and for all $n\ge N$ is precisely the same as being able to find $N$ such that $\sup_x |f_n(x) - f(x)| \le \epsilon $ for all $n\ge N$.
Explanation of 2: pointwise convergence allows you to choose $N$ based on $x$, if you want. Uniform convergence requires same $N$ for all $x$. A more restrictive definition implies a less restrictive one.