**READING: II §3 LUB3**
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II §3 LUB2

*begin quote ita***"**

*CHAPTER II* [:]
**
The Real Number System
**

Please get the book and read along. topic index contents

*
begin quote
ita-2-3-10-pg26
*
**"**

For any *x* there is an integer *n* such that *n**x**n*+1 **.**

**"***end quote*

I would like to consider two cases: (1) *x* is an integer, and (2) *x* is not an integer.

(1) *x* is an integer:

In this case, if we set *n* = *x* then the desired inequality becomes *n* = *x**n*+1 **.**

If we set *n* = *x-1* then the desired inequality becomes *n**x* = *n*+1 **.**

(2) *x* is not an integer:

In this case, *begin quote
ita-2-3-10-pg26***"** choose an integer *N*|*x*| **"***end quote* [which is permissible by LUB1] *begin quote ita-2-3-10-pg26***"** so that -*N**x**N* **"***end quote* [see the proof of this here]**.**

If *x*-*N*+1, then the desired inequality is obtained with *n* = -*N***.**

If *x* is not less than -*N*+1, then by the Order Preoperty and by the fact that *x* is not an integer,

it must be the case that -*N*+1*x***.**

Then if *x*-*N*+2, then the desired inequality is obtained with *n* = -*N*+1**.**

If *x* is not less than -*N*+2, then by the Order Preoperty and by the fact that *x* is not an integer,

it must be the case that -*N*+2*x***.**

Then if *x*-*N*+3, then the desired inequality is obtained with *n* = -*N*+2**.**

If *x* is not less than -*N*+3, then by the Order Preoperty and by the fact that *x* is not an integer,

it must be the case that -*N*+3*x***.**

This process must produce the desired inequality because, in the "worst" case where *x* is greater than all but the last element in the set *begin quote
ita-2-3-10-pg26***"** { -*N*, -*N*+1, ... ,0, 1, ..., *N*} **"***end quote*,

we have the desired inequality with *n* = *N*-1 **:** *N-1**x**N***.**

on to §3: LUB 4