Is the set of all Cauchy sequences of real numbers countable or uncountable?
In other words, is $S$ countable or uncountable, where $$S=\big\{\langle x_{n}\vert n\in\mathbb{N}\rangle\in\mathbb{R}^{\mathbb{N}} \, \big\vert \,\langle x_{n}\vert n\in\mathbb{N}\rangle \textrm{ is a Cauchy sequence} \big\}?$$
On
Consider the following facts:
Every convergent sequence is a Cauchy sequence.
For every real number $x$, there is a convergent sequence with $x$ as its limit. (There are in fact uncountably many such sequences, but we just need one for this argument to work.)
If two sequences converge to different limits, then they cannot be the same sequence.
There are uncountably many real numbers.
Let me know if you need help putting these observations together into an argument that there are uncountably many Cauchy sequences in $\mathbb{R}$.
On
Hint Given $f: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ a bijection, it is easy to construct an injective function from
$$(0,1)^\mathbb{N} \to [0,1] \,.$$
From here you can deduce that $\mathbb R^{\mathbb N}$ and $\mathbb R$ have the same cardinality. And you can easily squeeze your set in between.
$f(x) = (x, 0, 0, 0, \ldots)$ is an injective map from the set of real Cauchy sequences to $\mathbb{R}$.
If $x \in \mathbb{R}$ is written in base $11$ as $\ldots x_{-3}x_{-2}x_{-1}.x_1x_2x_3\ldots$, where $x_i$ is between $0$ and $10$, then $$ \begin{array}{ccc} f(x) = (&\ldots x_{-16}x_{-8}x_{-4}x_{-2}.x_2x_4x_8x_{16}\ldots, \\ &\ldots x_{-81}x_{-27}x_{-9}x_{-3}.x_3x_9x_{27}x_{81}\ldots, \\ &\ldots x_{-125}x_{-25}x_{-5}.x_5x_{25}x_{125}\ldots, \\ &\cdots \\ &\ldots x_{-p^3}x_{-p^2}x_{-p}.x_px_{p^2}x_{p^3}\ldots, \\ &\cdots &)\\ \end{array} $$ is a surjective map from $\mathbb{R}$ to the real set of Cauchy sequences. Note that while $x$ is in base $11$, the Cauchy sequence is in base 10. This is so as to avoid problems with equivalent decimal expansions for $x$. Note also that if $10$ appears in the decimal expansion for $x$, it is unclear what $f$ is defined as, but we don't care, because we just need a surjection.