I THINK this theorem in short words states that if a function is continuous in an interval [a,b] then it has a maximum. But I have a doubt with the demonstration when it supposes an absurd:(adsbygoogle = window.adsbygoogle || []).push({});

$$\begin{align}

& \text{THEOREM:} \\

& \text{ }f:[a,b]\to \mathbb{R}\text{ continuous} \\

& \text{then }\exists c\in [a,b]:f(c)\ge f(x)\forall x\in [a,b]\text{ } \\

& \text{ } \\

& \text{Dem}\text{. } \\

& \text{We know that }\exists \alpha =\sup f([a,b])\text{. } \\

& \text{Suppose as and absurd that }\forall x\in [a,b]\text{ }f(x)<\alpha \Rightarrow \alpha -f(x)>0 \\

& \text{ }...\text{continues}... \\

\end{align}$$

What I don't understand is... if alpha is sup then is the least upper bound, so wouldn't the absurd be that f(x) is GREATER than alpha, i.e. greater than a supposed number which we know is supremum?

Thanks!!

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# Doubt with a theorem that if f is cont. at interval it has max

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