I try to solve this problem in General Topology Stephen Willard :-
Show that
$|βN|\ge|βQ|$. [Consider any one-one map of $N$ onto $Q$ and use Theorem]
$|βQ| \ge|βR|$. [Consider the inclusion map of $Q$ onto $R$ and use Theorem]
$|βN| =|βQ| = |βR| = 2^\mathfrak{c}$. [N is C*-embedded in R ]
Definition 1: $βΧ$ is the Stone-Cech compactification of X.
Definition 2: A subset $A$ of a space $Τ$ is $C*$-embedded in $Τ$ iff every bounded continuous real-valued function on $A$ can be extended to $T$.
Theorem. If К is a compact Hausdorff space and $f: X \to K$ is
continuous, there is a continuous $F: βΧ \to K$ such that $F \circ e = f $
Is there exist a one-one map of $N$ onto $Q$ ? Which one ?
If there is one, what should i do then ??
I answered the first two in this answer For the last you only need that $|\beta \mathbb{R}| \ge |\beta \mathbb{N}|$ which is easy to see:
$X= \overline{\mathbb{N}}$( as a subset of $\beta \mathbb{R}$) is compact and has $\mathbb{N}$ as a dense subset and if $f: \mathbb{N} \to [0,1]$ is continuous, then we can extend $f$ to a map $f'$ from $\mathbb{R}$ to $[0,1]$ as $\mathbb{N}$ is $C^\ast$-embedded in the reals. Then we extend $f'$ to $\beta f' : \beta \mathbb{R} \to [0,1]$ and note that of course $\beta f' |_X$ is an extension of $f$ to $X$. So $X$ obeys the extension property that uniquely characterises $\beta \mathbb{N}$ so that $X =\overline{\mathbb{N}}^{(\beta \mathbb{R})} \simeq \beta \mathbb{N}$ and as $X \subseteq \beta \mathbb{R}$ we clearly have $|\beta \mathbb{N}| \le |\beta \mathbb{R}|$ as required.
In fact the above shows that if $A$ is $C^\ast$-embedded in $X$ then $\beta A \hookrightarrow \beta X$ (as $\overline{A}^{(\beta X)}$ in fact).
I believe that $|\beta \mathbb{N}|=2^{\mathfrak{c}}$ is shown in the text itself.