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Basic topology

Definition of component interval; Let S be an open subset of R. An open interval I is called a component itnerval of S if I is a subset of S and there is no open interval J =/= I such that I is a subset of J is a subset of S.

I am having trouble with the thereom below;

Every point of a non-empty open set S belongs to one and only one component interval of S,##Proof; Assume x is in S. Then x is in I where I is some open interval and I is a subset of S. The " largest" of these is;

(a(x),b(x)) where a(x) = inf{a | (a,x) is a subset of S_ and b(x) = sup{b | (x,b) is a subset of S}

I fail to see how (a(x),b(x)) is not S. Say S = (a,b) where a and b are real numbers. then a(x) clearly equals a and b(x) b.

Am I missing something here?