Intuition behind universal approximation theorem

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I can imagine a lot of math's theorems and laws, but I can not imagine the "process" universal approximation theorem is talking about.

Can you explain in simple words the proof of universal approximation theorem?

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The article A visual proof that neural nets can compute any function (by Michael Nielsen) gives a nice geometric intuition.

If you're interested, you can find the proof easily around, by following the Wikipedia references.

The linked book is generally good for understanding the basics of neural networks.