Does every monotone sequence converge or is it possible to show that there is a monotone sequence that does not converge? Any examples?
Any monotone real sequence has a general limit, either to a finite number (if the sequence is bounded from below or above, according as whether the seq. is monotone descending or ascending ,resp.), or to $\displaystyle \pm\infty$, again according as whether the seq. is mon. ascending or descending.
Tonio
An obvious example of an increasing sequence that does NOT converge is $\displaystyle a_n= n$. The sequence 1, 2, 3, 4, ... does not converge because it has no upper bound. Tonio would say that such a sequence has limit $\displaystyle \infty$ but it would be incorrect to say that such a sequence converges. It diverges to infinity.
The monotone convergence property is a "defining" property of the real numbers. Every increasing sequence of real numbers, with an upper bound, converges to a real number, every decreasing sequence of real numbers, with a lower bound, converges to a real number.
I say that is a "defining" property of the real numbers because it is equivalent to other properties such as the least upper bound property that are true for the real numbers, not the rational numbers.
For example, the sequence 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, 31.141592, etc., where every term is one more decimal place in the decimal expansion of $\displaystyle \pi$, is an increasing sequence of rational numbers, having 3.2, for example, as an upper bound but does not converge to a rational number. As a sequence of real numbers, of course, it converges to the irrational number $\displaystyle \pi$.