If $I_1,I_2,\dots$ are intervals of real numbers with lengths that sum to less than 1, then their union cannot be all of $[0,1]$. It is quite common for people to think this statement is more obvious than it actually is. (The "proof" is this: just translate the intervals so that the end point of $I_1$ is the beginning point of $I_2$, and so on, and that will clearly maximize the length of interval you can cover. The problem is that this argument works just as well in the rationals, where the conclusion is false.)
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