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LUB3: Integers Immediately Surrounding Any Real Number

*"*For any [real number] *x* there is an integer *n* such that .*"* *(see ita-2-3-13-pg26)*

*"*To prove this, choose an integer *N*>|*x*|, so that -*N*<*x*<*N*. The integers from -*N* to *N* form the finite set {-*N*,...,0,1,...*N*} and all we need do is take *n* to be the greatest of these that is less than or equal to *x*.*"* *(see ita-2-3-13-pg26)*

As a preliminary step towards proving Rosenlicht's first sentence immediately above, please see the page here on the negation of an inequality, by which I mean multiplying both sides of an inequality by -1. Note the similarity of the following proof with the *only if* proof in the subsection on *ε-neighborhoods* under *Absolute Values* in the *Order Property* section. Also recall that the *if and only if* logical statement was presented in the *Preliminary* part of the *Order Property* section.

For the 'take *n* to be the greatest of these' part of Rosenlicht's proof, please scroll down my first reading of LUB3 here.

Converted by *Mathematica*
December 4, 2007