Real Numbers are Uncountable/Cantor's First Proof

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Theorem

The set of real numbers $\R$ is uncountably infinite.


Proof

We prove the equivalent result that every sequence $\sequence {x_k}_{k \mathop \in \N}$ omits at least one $x \in \R$.


Let $\sequence {x_k}_{k \mathop \in \N}$ be a sequence of distinct real numbers.

Let a sequence of closed real intervals $\sequence {I_n}$ be defined as follows:

Let:

$a_k = \min \set {x_k, x_{k + 1} }$
$b_k = \max \set {x_k, x_{k + 1} }$

and:

$I_k = \closedint {a_k} {b_k}$

Since the terms of $\sequence {x_k}_{k \mathop \in \N}$ are distinct, $a_k \ne b_k$.

Thus $I_k$ is not a singleton.

Let:

$I_{n - 1} = \closedint {a_{n - 1} } {b_{n - 1} }$

It can be assumed that infinitely many of the $x_k$ lie inside $I_{n - 1}$.

Otherwise the proof is complete.


Let $y$ and $z$ be the first two such terms of $\sequence {x_k}_{k \mathop \in \N}$.

Let:

$a_n = \min \set {y, z}$
$b_n = \max \set {y, z}$
$I_n = \closedint {a_n} {b_n}$


Thus we have sequences:

$\sequence {a_k}_{k \mathop \in \N}$
$\sequence {b_k}_{k \mathop \in \N}$

with:

$ a_1 < a_2 < \cdots < b_2 < b_1$

So $\sequence {a_k}_{k \mathop \in \N}$ and $\sequence {b_k}_{k \mathop \in \N}$ are monotone, and bounded above and bounded below respectively.

Therefore by the Monotone Convergence Theorem (Real Analysis) both are convergent.

Let:

$\displaystyle A = \lim_{k \mathop \to \infty} a_k$
$\displaystyle B = \lim_{k \mathop \to \infty} b_k$

Clearly we have $A \le B$.

So:

$\closedint A B \ne \O$

Let $h \in \closedint A B$.

Then:

$h \ne a_k, b_k$ for all $k$.


We claim that $h \ne x_k$ for all $k$.

Suppose that $h = x_k$ for some $k$.

Then there are only finitely many points in the sequence before $h$ occurs.



Therefore only finitely many of the $a_k$ precede $h$.

Let $a_d$ be the last element of the sequence $\sequence {a_k}_{k \mathop \in \N}$ preceding $h$.

We defined $a_{d + 1}$, $b_{d + 1}$ to be interior points of $I_d$, and also $h \in I_{d + 1}$ by the definition of $h$.



Therefore $a_{d + 1}$ must precede $h$ in the sequence, for the sequence is monotone increasing.

This is a contradiction.



$\blacksquare$


Historical Note

This proof was first demonstrated by Georg Cantor.