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Suppose that $\alpha =\sup S$ and $\alpha \notin S$. Then prove that $\alpha$ is an accumulation point of $S$ and $S$ is infinite.

I know that

Let $S\subseteq R$ be nonempty and bounded above, and let $\beta=\sup S$ and $\beta\notin S$. Prove that for each $\epsilon>0$ the set $\{x\in S:x>\beta−\epsilon\}$ is infinite.

How use this result to prove the above problem?

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    I suppose $S$ is a subset of the real numbers? What is your definition of supremum? The least upper bound? Hint: If $S$ is finite, then by induction there is a maximum. If $\alpha$ is not a accumulation point and not in $S$, in particular it is not isolated, can you produce a smaller upper bound? – 2017-02-09
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    I tried to edit the format of your question but it is not so clear what you mean in the second part. – 2017-02-09

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If $\alpha=\sup S$, then for each $\varepsilon>0$, there is $s\in S$ with $\alpha-\varepsilon

Does a finite set have accumulation points?