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begin quote ita"
CHAPTER II [:] The Real Number System [...] 3. THE LEAST UPPER BOUND PROPERTY. "end quote
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LUB1:      

begin quote ita-2-3-10-pg26 "
For any real number   x,   there is an integer   n,   such that   nx.
"end quote

It seems to me that we already know from defining the natural numbers that there are arbitrarily large integers. Therefore, "existence of arbitrarily large integers" seems to be not precisely what is being asserted here. Hence the down arrow above replaces the statement of title that is usually present at the top of these pages.

Also (8 May 2012):

REFERENCE: A Primer of Infinitesimal Analysis, by J. L. Bell, Cambridge University Press, 1998. Hereafter: Bell_APoIA. This book is based on the mathematical subject of category theory - which now also has been added to my list for future study along with non-standard analysis.

As I was reading Bell_APoIA for application to my studies in classical tensor analysis, I learned (page 106) that LUB1 above is called the Archimedean principle. It is fascinating to note (also page 106) that this principle does not hold in some models of smooth infinitesimal analysis. Also, in conjunction with what has just been said, I see (page 13) that the reductio ad absurdum proof by contradiction is to be avoided. I note this with some satisfaction after pausing here as I did - being leary of the proof by contradiction.

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