Example of a convergent pointwise sequence of function and not uniformly convergent

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  • We say that a sequence $f_n(x)$ is pointwise convergent to $f(x)$ on the set $E$ when for every $x\in E$ and $\epsilon >0$ there exists an integer $N$ such that the distance between $f_n(x)$ and $f(x)$ is smaller than $\epsilon$.
  • We say that a sequence $f_n$ is uniformly convergent to $f$ on the set $E$ if for all $\epsilon >0$ there exists an integer $N$ such that the distance between $f_n$ and $f$ is smaller than $\epsilon$. The main difference between the aforementioned definitions is that $N$ dependant on $x$ in the context of pointwise convergence, that is for each $x$ in $E$ there exists a specific $N$ associated to $x$ while in the case of uniform convergence there exists a universal $N$ for which the $f_n(x)$ converges to $f(x)$ uniformly with respect to $x$. Can you please give me an example of sequence of function that converges pointwise and not unifomly to $f(x)$. Thanks in advance