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Question

Explain what it means to find a "counterexample" to a universally quantified mathematical statement. Find a counterexample for the following: "for all ε>0, there exists a δ>0 such that if x is a real number with 0 <|x −a| <δ, then |1/x| <ε.” Justify why your example is indeed a counterexample.

Answer 1

The statement "for all ε>0, there exists a δ>0 such that if x is a real number with 0 <|x −a| <δ, then |1/x| <ε.” could be written in the logic language as "∀ε>0 ∃δ>0 0 < |x −a| < δ → |1/x| < ε"

To find a counterexample we need to prove that:

There exists ε>0 such that for any δ>0 there exists such value of x for that 0 < |x −a| < δ and |1/x| ≥ ε.

(In the logic language "∃ε>0 ∀δ>0 ∃x 0 < |x −a| < δ ∧ |1/x| ≥ ε")

The original statement has a variable a. Find the counterexample for any value of a.

Let a ≠ 0. Select ε = 1/|a|. Then for any δ > 0 there exists value of x for that 0 < |x −a| < δ and |x| < |a|.

For this x |1/x| > 1/|a| = ε.

Let a = 0. Then for any ε > 0 and any δ > 0 there exist a value of x such that 0 < |x| < δ and |x| < 1/ε.

For this x |1/x| > ε.

Answer 2

A counterexample occurs when a=0 because near 0 1/x is large and, in fact, the smaller the value of delta the larger 1/x grows.

In general a counterexample is an example which meets the conditions of the hypothesis (if statement) but does NOT meet the criteria in the conclusion (then statement). Providing a counterexample to a proposition proves that the proposition is false.

P.S.

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