## 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) limx→c f(x) = L.

(ii) The ε, δ condition for limx→c f(x) = L is true.

PROOF We first assume the ε, δ condition and prove that

limx→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

limx→c f(x) = L.

For the other half of the proof we assume that

limx→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 δ1 > 0 be a positive infinitesimal. By the Transfer Principle, Equation (1) holds for δ1 Therefore x1 = x(δ1) is infinitely close but not

equal to c. But since

|f(x1) - L| ≥ε

and ε is a positive real number, f(x1) is not infinitely close to L. This contradicts the equation

limx→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.

Last Update: 2006-11-05