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## Homework Statement

Suppose that a sequence {p}->p, and {q}->q. Show that if p_n < q_n for infinitely many n, then p<q.

## Homework Equations

## The Attempt at a Solution

I'm sure I know why this is true intuitively. The convergence guarantees that all but a finite number of p_n are within some distance of p, and all but a finite number of q_n are within some distance of q, for large enough n this distance is arbitrarily small, therefore, if an infinite number of p_n < q_n, then an infinite number of p_n < q_n when in some neighborhood of p which can be made as small as we like; hence in the limit, p<q.Now, for actual rigor... Does it suffice to say that the d(p, p_n) < r, d(q, q_n) < r for some n if we take n > max (N, N') (where N and N' are the natural numbers such that the above inequalities hold in each case). Therefore in any interval around p and q, there are an infinite number of p<q, ect.

I know this should be an epsilon argument, but I don't know how to make it less "qualitative"

This is my problem with math. Urgh. Help please.

Edit: Yeah, I should've said < or =, but there is = on the keyboard, so I let that slide.

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