Proposition: Equivalence of Definitions of Limits

• If f is any function with domain D in R, and c closure(D) then the following are equivalent:
  1. For any sequence {} in D that converges to c the sequence f() converges to L
  2. given any > 0 there exists a > 0 such if x closure(D) and | x - c | < then | f(x) - L | <

Proof:

Suppose the first condition is true, but the second condition fails. Then there exists an > 0 such that there is no > 0 with the property that if | x - c | < then | f(x) - L | < . Therefore, if we let = 1 / n, then for each n we can produce a number with

• | - c | < but | f() - L |

But then the sequence { } converges to c, but the sequence f() does not converge to L. That is contrary to the first condition being true, and hence we have proved by contradiction that the first condition implies the second.

Suppose the second condition is true. Let c be some number in closure(D) and pick any > 0. There exists a number > 0 such that

• whenever | x - c | < then | f(x) - L | <

Take any sequence { } in D that converges to c. Then there is an integer N such that

• | - c | < for n > N

But then, by assumption of the second condition,

• | f() - L | < for n > N

But that is the definition of the sequence { f() } converging to L, as required.