Uniform and point wise convergent What is the difference between point wise convergence and uniform convergence?
Can you explain the answer geometrically? 
 A: You can easily visualize the difference form functions from $\mathbb{R}$ to $\mathbb{R}$ by looking at their graph:


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*In pointwise convergence you have that for every $x_{0}$ the point $(x_{0},f_{n}(x_{0}))$ "get closer" to the point $(x_{0},f(x_{0}))$ along the vertical.

*In uniform convergence you have something stronger: it morally means that you can take a "tube" around the graph of $f(x)$ in $\mathbb{R}\times\mathbb{R}$, small as you like, and the graph of $f_{n}(x)$ will always be in that "tube" for large enough $n$.

A: We say a function $f_n$ converges pointwise to $f$, if for a fixed $x$, as $n \rightarrow \infty$, $f_n(x) \rightarrow f(x)$.
For uniform convergence, it is required that for all $x$ as $n \rightarrow \infty$ $f_n \rightarrow f$. You can think of this as a 'tube' around the function: as $n \rightarrow \infty$, $f_n$ is contained within this 'tube' about $f$ (in fact it is a $\epsilon$ distance surrounding $f$.
As an example, consider the function $f_n(x) = x^n$, for $x = 1$, $f_n(1) \rightarrow 1$ as $n \rightarrow \infty$, so $f_n$ converges pointwise to $1$ for $x = 1$. 
However, for say $x=2$, $f_n(2) \rightarrow \infty$, as $n \rightarrow \infty$, so it is not uniformly convergent, as it does not converge to the same 'thing' for all $x$.
A: You may think in this way: Let say we have a tube, in fact this should be range, then uniform convergence simply requires the diameter of the tube is sort of uniform at infinity.
