There are all these examples surrounding fixed point theorems.
The following is somehow a cliche. Take a sheet of paper, crunch it, and put it on top of its original position. Then there is a point that lies on the vertical of its previous position. This illustrates the fixed point theorem for contractions in Banach spaces.
There are also a lot of examples in probability theory.
Here is one related to the harmonic series. In your youth, you may have collected cards depicting soccer players, martians, whatever. There is a finite number of cards to collect, say n. Each packet of corn flakes comes with one of them, at random. And of course you want your mom to buy this precise brand of flakes so as to get the whole collection. May be you have wondered what is the average number of packets she should buy so that you can complete the collection. The answer is
$$
n sumlimits_{k=1}^n {1 over k}
$$
This is asymptotic to $nlog n + ngamma+ 1/2$, where $gamma$ is the Euler constant. This is the simplest mundane example I know involving that constant. So for example, if there are $n=150$ cards to collect, you need to buy an average of 519 cards.
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