Density Property Of The Real Numbers

#4. Density property
The density property tells us that we can always find another real number that lies between any two real numbers. For example, between 5.61 and 5.62, there is 5.611, 5.612,5.613 and so forth.
Between 5.612 and 5.613, there is 5.6121, 5.6122 ... and an endless list of other numbers!

SECTION 6.5: THE DENSITY PROPERTY OF REAL NUMBERS
Between the numbers 2 and 3, there isanother number 2.1. In between the two numbers .00000000001 and .00000000002, there is another number .000000000017.
The "density property of real numbers" states that between any two real numbersthere is another number. This property ultimately leads us to the following statement: between any two real numbers there are an infinite number of numbers!
DENSITY OF THE REALS
  The false picture ofthe reals as a “row of points” suggests that to the right of each real number is the “next” real number  That is wrong:  The points on the real line are dense, meaning that the real line has nogaps.   More precisely:
 
Between any two distinct points on the real line
is an infinite number of other points.  
 
The proof is just like that for the density of the rationals:  If r and s are twodifferent real numbers with , then  is between them:
                                                          (WA)

DENSIDAD Dados a; b 2 R si a > b entonces existen un elemento x 2 R
tal que a> x y x > b:

 Lección 1.9.   |    Densidad de los Números Racionales y de los Números Irracionales en R |
Dados dos números reales diferentes  y , su promedio  esta comprendido entre  y .Por lo tanto, entre dos números reales sin importar lo cercano que se encuentren, hay una infinidad de números reales. Esto implica que dado un número real cualquiera  no tienen sentido expresionestales como " el número real siguiente a " o " el número real anterior a ".
Usando nuestra caracterización de los números reales como expresiones decimales, podemos refinar el resultado anterior y...