Originally Posted by

**larson** Use Theorem 3 to show that the sequence 0.7, 0.77, 0.777, 0.7777, 0.77777, 0.777777, 0.7777777, ... has a limit.

Theorem 3 is ...

Suppose that the sequence $\displaystyle a_{k}$ is nondecresing and bounded above by a number A. That is,

$\displaystyle a_{1}$ <, $\displaystyle a_{2}$ < $\displaystyle a_{3}$ < A

Then $\displaystyle a_{k}$ converges to some finite limit a, with a < A. Similarly, if $\displaystyle b_{k}$ is nonincreasing and bounded below by a number B, then $\displaystyle b_{k}$ converges to a finite limit b > B.

edit: How do I do the less than or equal to sign in LaTeX?