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In Munkres Topology, page $172$ there is the theorem: "Let $X$ be a simply ordered set having the least upper bound property. In the order topology, each closed interval in $X$ is compact"

In proof says: "if $x$ is a point of $[a,b]$ different from $b$, then there is a point $y>x$ of $[a,b]$ such that the interval $[x,y]$ can be covered by at most two elements of $A$ ($A$ is a covering of $[a,b]$) and he proofs that saying: "if $x$ has an immediate successor in $X$ ,let $y$ this immediate successor $[x,y]$ consists of the two points $x$ and $y$, so that it can be covered by at most two elements of $A$"

Can anyone explain this to me? Why at most two elements of $A$?

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We know that $[x, y] = \{x, y\}$ (otherwise, if there is some $z \in (x, y)$, then $z$ would be a more immediate successor of $x$ than $y$, a contradiction). Since $\mathcal A$ covers $\{x, y\}$, we know by definition that there exist some sets $S_x, S_y \in \mathcal A$ such that $x \in S_x$ and $y \in S_y$. Thus, since $\{S_x, S_y\}$ covers $\{x, y\}$, we know that $[x, y]$ can be covered by two elements of $\mathcal A$ (it's possible that we could get away with just one element, but we're only trying to find a finite subcovering; it's okay if we use up an extra element in our covering - there's no need to be stingy with our elements).

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    i cannnot understand why two elements maximum. For example the set (a,b) can also cover [x,y] or i am wrong? – 2017-01-16
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    It's possible to cover it with more than two elements. But the statement in the proof is that it **can** be covered with two elements. Remember that our goal here is to find an upper bound on the number of elements in our subcovering (we want to get away with only using a finite number of elements). – 2017-01-16
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    i keep our goal on mind but when Munkres says at most two i thought that he means only two and no more so i was so confused. – 2017-01-16
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    It means that at most two are *required*, not that at most two are *allowed*. He's trying to keep track of how many are needed in total. – 2017-01-16