# Greatest lower bound proof

## Homework Statement

Use the Archimedean property of $\mathbb{R}$ to prove that
the greatest lower bound of ${\frac{1}{n}:n\in\mathbb{N}}=0$
the archimedean principle says that for any number y there is a natural number
such that 1/n<y for y>0

## The Attempt at a Solution

since all of our numbers in our set are positive. I could pick a real number as close to zero as I wanted but there would still be a natural such that 1/n is smaller than the real I picked, there zero is the greatest lower bound of the set.

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Dick
Homework Helper
Well, yes, that would be true. What's your question?

Im guessing my statement isn't enough to prove it. Or should I assume that there is a real just to the right of zero and claim that this is the greater lower bound, but then there is always another number 1/n that is smaller and since all 1/n are in the set, that zero has to be the greatest lower bound.

Last edited:
Dick