begin quote ita"
CHAPTER II [:] The Real Number System [...] §3. THE LEAST UPPER BOUND PROPERTY. "end quote
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LUB3: NEAREST INTEGERS TO ANY REAL NUMBER

begin quote ita-2-3-10-pg26 "
For any   x   there is an integer   n   such that   nxn+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 = xn+1 .
If we set   n = x-1   then the desired inequality becomes   nx = 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   -NxN "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+1x.

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+2x.

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+3x.

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-1xN.

on to §3: LUB 4