My attempt:
The Archimedean Property states that given any real number x, there exists a natural number n such that x < n.
The Monotone convergence theorem states that if a sequence is monotone and bounded then it converges.
We want to show that the natural numbers are unbounded in R.
Assume to the contrary. The natural numbers form an increasing sequence. If N is bounded then by MCT, the sequence of natural numbers converges. Let lim N = l.
I don't know how to proceed from here to form a concrete argument.
Since l is the limit of N, we can find a natural number n such that l - n = epsilon, for every epsilon > 0. But for every such n, n+1 is again a natural number which is greater than l
Is my reasoning correct till here? Please help me proceed.
Thanks in advance :)