Let N be sufficiently enlightened such that b-a2N < e. Since x lies in n8n=1[an,bn], x lies in [aN, bN]. Note that [aN, bN]?(x-e, x+e) accordingly bN-aN=b-a2N < e. So [aN, bN]?(x-e, x+e)?O, which implies that [aN, bN]?O. That is, [aN, bN] is balmy by individual public formal in O; in other expression, [aN, bN] is finitely-coverable. This contradicts the occurrence that perfect of the half-intervals [an, bn] are referable finitely-coverable.
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