Talk:Intermediate value theorem

I'm not a mathematician, but there's a bit of this proof that seems wrong. It goes

"Suppose first that f (c) > u."

I don't see how we can suppose this since by definition

    c = sup {x in [a, b] : f(x) ≤ u}. 

which I understand as meaning that c is the largest number of those real numbers x such that f(x) is less or equal to u. If so, how could f(c) be greater than u??