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 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

lim_x→c f(x) = L.

We conclude that the ε, δ condition must be true after all.