We now turn to the question whether approximates when approximates x. This is the same as asking: suppose is a sequence of real numbers converging to x, does converges to ?

**Theorem.**

If is continuous at if and only if converges to whenever converges to x, that is,

.

**Proof.**

Suppose f is continuous at x and converges to x. By continuity, for every , there exists such that whenever . Since converges to x, for this , we can find an such that for .

So for as .

Conversely, suppose converges to when converges to x. We have to show that f is continuous at x. Suppose f is not continuous at x. That is to say, there is an such that however small we may choose, there will be a y satisfying yet . So for every n, let be such a number for which and . Now, we see that the sequence converges to x. But does not converge to f(x) violating our hypothesis. So f must be continuous at x. QED.

More later,

Nalin Pithwa