Intermediate Value Theorem
As another illustration of the power of Robinson's approach, we present a short proof of the intermediate value theorem (Bolzano's theorem) using infinitesimals.
Let f be a continuous function on such that f(a)<0 while f(b)>0. Then there exists a point c in such that f(c)=0.
The proof proceeds as follows. Let N be an infinite hyperinteger. Consider a partition of into N intervals of equal length, with partition points xi as i runs from 0 to N. Consider the collection I of indices such that f(xi)>0. Let i0 be the least element in I (such an element exists by the transfer principle, as I is a hyperfinite set; see non-standard analysis). Then the real number
is the desired zero of f. Such a proof reduces the quantifier complexity of a standard proof of the IVT.
Read more about this topic: Non-standard Calculus
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