Intermediate value theorem

The intermediate value theorem (IVT) tells us that for a function that is continuous on some interval $[a,b]$ and there is some value $z\in \mathbb{R}$ that is between $f(a)$ and $f(b)$, then there is a number $c\in [a,b]$ such that $f(c)=z$.

This seems like a pretty obvious idea, but can lead to solid foundations for results in real analysis. I think a cool way to think about it is that at some point your height was exactly $\frac{\pi }{3}$ metres tall to infinite precision.