'Obvious' theorems that are actually false

Theorem (false):

One can arbitrarily rearrange the terms in a convergent series without changing its value.

A shape with finite volume must have finite surface area.

I wish I'd thought of this yesterday, when the question was fresh, because it's astounding. Suppose $A$ and $B$ are playing the following game: $A$ chooses two different numbers, via some method not known to $B$, writes them on slips of paper, and puts the slips in a hat.

$B$ draws one of the slips at random and examines its number. She then predicts whether it is the larger of the two numbers.

If $B$ simply flips a coin to decide her prediction, she will be correct half the time.

Obviously, there is no method that can do better than the coin flip.

But there is such a method, described in Thomas M. Cover “Pick the largest number”Open Problems in Communication and Computation Springer-Verlag, 1987, p152.

which I described briefly here, and in detail here.