Can we assume that you are working in therealnumber system? That statement is not true, for example, in the rational numbers.

Now, I presume you know that anynon-decreasingsequence of real numbers, having anupperbound, converges (and, of course, anynon-increasingsequence of real numbers, having alowerbound, converges). That's a defining property of the real numbers.

What you need to do is show that every sequence contains amonotonesubsequence. Here's a hint: Let S be the set of indices, i, such that if j> i, then . Consider two cases: S is infinite or S is either empty or finite.