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24 April 2024
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N 07:38 | Principle of Open Induction for Real Numbers 2 changes history +2,644 [Prime.mover; CircuitCraft] | |||
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07:38 (cur | prev) +36 Prime.mover talk contribs | ||||
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02:53 (cur | prev) +2,608 CircuitCraft talk contribs (Created page with "== Theorem == Let $a < b$ be real numbers. Let $S$ be an open set of real numbers. Suppose that, for every $x \in \closedint a b$ such that: :$\hointr a x \subseteq S$ it also holds that: :$x \in S$ Then, $\closedint a b \subseteq S$. == Proof == {{AimForCont}} there exists some $x \in \closedint a b$ such that: :$x \notin S$ Let: :$T := \closedint a b \setminus S$ be the set o...") |