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The Real Line

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Order on the Real Numbers

Let IR be the field of real numbers, we define order on IR by the relation "less than" or " < ", such that, &forall. ~a, ~b, ~c, &in. &reals.:

  1. if ~a != ~b then either ~a < ~b or ~b < ~a
  2. ~a < ~b and ~b < ~c => ~a < ~c
  3. ~a < ~b => ~a + ~c < ~b + ~c, any ~c &in. &reals.
  4. 0 < ~a and 0 < b => 0 < ~a~b

Write _ ~a =< ~b _ ("less than or equal to") _ if ~a < ~b or ~a = ~b
also, write _ ~a > ~b _ ("greater than") _ <=> ~b < ~a
and _ ~a >= ~b _ ("greater than or equal to ") _ if ~a > ~b or ~a = ~b

We can deduce the following:

  1. ~a > 0 _ then _ -~a < 0
  2. ~a > ~b _ and _ ~c > ~d _ => _ ~a + ~c > ~b + ~d
  3. ~a > ~b _ and _ ~c > 0 _ => _ ~a~c >
  4. ~a > ~b _ and _ ~c < ~d _ => _ ~a + ~c < ~b + ~d
Proof:
  1. ~a > 0 => ~a + (-~a) > 0 + (-~a) => 0 > -~a
  2. ~a > ~b => ~a + ~c > ~b + ~c , and ~c > ~d => ~b + ~c > ~b + ~d , so ~a + ~c > ~b + ~d (by ii.)
  3. ~a > ~b => ~a - ~b &gt 0 so (~a - ~b)~c = ~a~c - ~b~c > 0 (by iv) => ~a~c > ~b~c
  4. (~a - ~b) > 0, -~c > 0 (by v.) so -~c(~a - ~b) = ~b~c - ~a~c > 0 => ~b~c > ~a~c

Well Ordering Axiom on the Natural Numbers

&naturals. &subset. &reals., IN = {1, 2, 3,...} the set of Natural Numbers or Positive Integers.

Axiom: A non-empty set of positive integers has a least member. i.e:

~S &subseteq. &naturals. , _ ~S != &empty. , _ then &exist. ~x &in. ~S such that _ ~x =< ~y , _ &forall. ~y &in. ~S

Note that ~x is unique (for ~S) for suppose ~x, ~x' least members, then ~x =< ~x' and ~x' =< ~x, so ~x = ~x'

Least Positive Integer

There is no integer between 0 and 1.

Proof:

For any ~a &in. &reals. , _ 0 =< 1 , _ then _ 0 =< ~a^2 =< ~a =< 1.
If ~S = \{~n &in. &naturals. | ~n < 1 \} != &empty. , _ then ~S has a smallest member, _ ~m say , _ 0 < ~m < 1 _ => _ 0 =< ~m^2 =< ~m =< 1, _ => _ ~m^2 &in. ~S , _ Contradiction.

Principle of Induction

~S &subseteq. &naturals. , _ if _ 1 &in. ~S , _ and _ ~n &in. ~S => ~n + 1 &in. ~S , _ then ~S = &naturals.

Proof:

Suppose ~S != &naturals. , _ then if ~X = &naturals. #\ ~S, _ ~X != &empty., and &therefore. has a least member, ~m say, ~m != 1. So ~m - 1 &in. &naturals., but ~m - 1 &nin. ~X, so ~m - 1 &in. ~S => (~m - 1) + 1 &in. ~S by inductive hypothesis. So ~m &in. ~S. Contradiction.

Archimedes Axiom

For any ~x, ~y &in. &reals. , _ where _ ~x > 0, ~y > 0, _ &exist. ~n &in. &naturals. such that _ ~n~x > ~y.

Corollaries:

  1. given _ &epsilon. > 0 , _ &epsilon. &in. &reals. , _ &exist. ~n &in. &naturals. , _ such that _ 1/~n < &epsilon.. _ [i.e. _ &exist. ~n &in. &naturals., _ such that _ ~n&epsilon. > 1]
  2. &integers. &subset. &reals., _ &integers. = \{...-2, -1, 0, 1, 2 ... \}. _ Given _ ~x &in. &reals. , _ &exist. ~z_1, ~z_2 &in. &integers. _ such that _ ~z_1 < ~x < ~z_2.

Proof: (of ii.)

  1. ~x > 0: _ _ &exist. ~z_2 > ~x and let _ ~z_1 = 0.
  2. ~x < 0: _ _ ~z_2 = 0, _ now -~x > 0 _ so _ &exist. ~n _ such that _ ~n > -~x , _ let _ ~z_1 = -~n , _ ~z_1 < ~x.
  3. ~x = 0: _ _ -1 < ~x < 1.

Metric on IR

Define the absolute value of ~x &in. &reals. as

| ~x | = ~x , _ if _ ~x >= 0

| ~x | = -~x , _ if _ ~x < 0

We have: | ~x | >= 0, _ &forall. ~x &in. &reals., _ and _ | -~x | = | ~x | .

Define the usual metric on IR : _ ~x, ~y &in. &reals. , _ d(~x, ~y) = | ~x - ~y |

we have:

  1. d(~x, ~y) >= 0, _ &forall. ~x, ~y &in. &reals.
  2. d(~x, ~y) &eq. 0 _ <=> _ ~x &eq. ~y
  3. d(~x, ~y) &eq. d( ~y, ~x), _ &forall. ~x, ~y &in. &reals.
  4. d(~x, ~y) < d(~x, ~z) + d( ~y, ~z), _ &forall. ~x, ~y, ~z &in. &reals. _ (triangle inequality.)

Rational Numbers arbitrarily close to Real Numbers

Q &subset. &reals. _ - _ the set of rational numbers. _ ~q &in. Q _ => _ ~q = ~a/~b ; _ ~a, ~b &in. &integers.

There are rational numbers arbitrarily close to any real number.
i.e. _ for _ ~y &in. &reals. , _ given _ &epsilon. > 0 , _ &exist. _ ~p/~q &in. Q _ such that _ | ~y - ~p/~q | < &epsilon.

Proof:

  • if _ ~y >= 0 , _ &exist. ~q &in. &naturals. _ such that _ 1/~q < &epsilon. _ (corollary i to Archimedes Axiom),
    and _ &exist. ~n &in. &naturals. _ such that _ ~n(1/~q) > ~y _ (Archimedes Axiom).
    Let _ ~p _ = _ min \{ ~n &in. &naturals. | ~n(1/~q) > ~y\} , _ then _ (~p - 1)/~q < ~y < ~p/~q , _ so _ | ~y - ~p/~q | =< 1/~q < &epsilon.
  • Similarly for ~y < 0

Square Root of 2 is not Rational

&sqrt.2 &nin. Q

Proof:

Suppose _ (~p/~q)^2 = 2, _ and _ ~p/~q _ is in lowest terms. _ ~p^2 = 2~q^2 _ => _ ~p^2 is even , _ ~p = 2~r _ say, _ then _ (2~r)^2 = 2~q^2 , _ or _ 2~r^2 = ~q^2 _ => _ ~q^2 is even , _ so both ~p and ~q are even _ => _ ~p/~q is not in lowest terms. _ Contradiction.