In applying Newton's Method to solve f(x) = 0, one can usually tell by simply looking at

Question:

In applying Newton's Method to solve f(x) = 0, one can usually tell by simply looking at the numbers xl, r2, x3, ..... whether the sequence is converging. But even if it converges, say to I, can we be sure that is a solution? Show that the answer is yes provided f and f' are continuous at and f'(x) ≠ 0.