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Home Limits, Analytic Geometry, and Approximations The ε, δ Condition for Limits Equivalence of the ε, δ Condition to the Infinitesimal Definition of a Limit  
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Equivalence of the ε, δ Condition to the Infinitesimal Definition of a Limit
We conclude this section with the proof that the ε, δ condition is equivalent to the infinitesimal definition of a limit. THEOREM 1 Let f be defined in some deleted neighborhood of c. Then the following are equivalent: (i) lim_{x→c} f(x) = L. (ii) The ε, δ condition for lim_{x→c} f(x) = L is true. PROOF We first assume the ε, δ condition and prove that lim_{x→c} f(x) = L. Let x be any hyperreal number which is infinitely close but not equal to c. To prove that f(x) is infinitely close to L we must show that for every real ε > 0, f(x)  L  < ε. Let ε be any positive real number, and let δ > 0 be the corresponding number in the ε, δ condition. Since x is infinitely close to c and δ > 0 is real, we have 0 < x  c < δ. By the ε, δ condition and the Transfer Principle, f(x)  L < ε. We conclude that f(x) is infinitely close to L. This proves that lim_{x→c} f(x) = L. For the other half of the proof we assume that lim_{x→c} f(x) = L, and prove the ε, δ condition. This will be done by an indirect proof. Assume that the ε, δ condition is false for some real number ε > 0. That means that for every real δ > 0 there is a real number x = x(δ) such that (1) x ≠ c, x  c < δ, f(x)  L ≥ ε. Now let δ_{1} > 0 be a positive infinitesimal. By the Transfer Principle, Equation (1) holds for δ_{1} Therefore x_{1} = x(δ_{1}) is infinitely close but not equal to c. But since f(x_{1})  L ≥ε and ε is a positive real number, f(x_{1}) is not infinitely close to L. This contradicts the equation lim_{x→c} f(x) = L. We conclude that the ε, δ condition must be true after all. The theorem is also true for the other types of limits. The concept of continuity can be described in terms of limits, as we saw in Section 3.4. Therefore continuity can be defined in terms of the real number system only.


Home Limits, Analytic Geometry, and Approximations The ε, δ Condition for Limits Equivalence of the ε, δ Condition to the Infinitesimal Definition of a Limit 