Are there mathematical theorems that sound trivial and are obviously true, but are really tough to rigorously show?

For example I stumbled across this [question](https://math.stackexchange.com/questions/78077/intersection-of-two-curves-in-a-square). The author asks how to show that two continuous curves that connect the points $(0,0)$ with $(1,1)$ and $(0,1)$ with $(1,0)$ always have an intersection (the curves are restricted to $[0,1]\times[0,1]$). I asked a few persons and they all agreed that it is clear to be true from a geometrical point of view, but the actual proof is really tricky (especially if you do not use topology).
This surprised me and also frightened me a bit, because I thought that trivial theorems always have trivial proofs (otherwise it would be difficult to learn mathematics). Are there more such examples or even stronger ones?