Obvious, but Difficult
This is a fun conversation on MathOverlflow.net about famous examples of theorems in mathematics that are “obvious” but very difficult to prove.
For example, the Jordan Curve Theorem essentially states that any closed curve in the plane divides the plane into an “inside” and an “outside”. Obvious, right? But very difficult to prove.
The Isoperimetric Theorem is another good example. This theorem basically says that the most efficient way to surround area in the plane is with a circle. Again, easier to believe than to prove.
And one of the responders notes that, after taking several hundred pages in their Principia Mathematica to prove that , Russell and Whitehead note that the proposition “is occasionally useful”.
“Obvious” is one of the most dangerous words in mathematics!