NOTE: I am working with a discrete topology on $\mathbb N$ and the Stone-Čech compactification of this space. But the same proof works for any discrete space.$\newcommand{\N}{\mathbb N}\newcommand{\Filt}[1]{\mathcal{#1}}\newcommand{\UA}[1]{{\widehat{#1}}}\newcommand{\prin}[1]{{{#1}^*}} \newcommand{\Obr}[2]{{#1}[#2]}\newcommand{\Invobr}[2]{{#1}^{-1}[#2]} \newcommand{\ol}[1]{\overline{#1}}\newcommand{\FF}{{\mathcal{F}}}\newcommand{\GG}{{\Filt{G}}}
\newcommand{\Flim}{\operatorname{\mathcal{F}-lim}}\newcommand{\Glim}{\operatorname{\mathcal{G}-lim}}
\newcommand{\Zobr}[3]{{#1}\colon{#2}\to{#3}}\newcommand{\sm}{\setminus} \newcommand{\emps}{\emptyset}$
Let $\beta\N$ be the set of all ultrafilters on $\N$. The set of free ultrafilters is denoted by $\beta\N^*$.
The principal ultrafilter determined by $n$ will be denoted by $\prin n$.
Notice that we have
$$\begin{gather*}
\UA {A\cap B}=\UA A \cap \UA B \\
\UA {A\cup B}=\UA A \cup \UA B \\
\UA {\N\sm A}=\beta\N\sm\UA{A}
\end{gather*}$$
From the first property we see that the set $\{\UA A; A\subseteq\N\}$ is a base for a topology on $\beta\N$.
Theorem. The topology on $\beta\N$ generated by $\{\UA A; A\subseteq\N\}$ is compact and Hausdorff. This space is zero-dimensional.
The map $\Zobr e{\N}{\beta\N}$ which maps $n$ to $n^*$
is a dense embedding if $\N$ is endowed with the discrete topology. Moreover, for each sequence $\Zobr x{\N}K$ to a compact Hausdorff space $K$ there is a unique continuous extension $\Zobr{\ol x}{\beta\N}K$ such that $\ol x\circ e=x$. $($This extension can be expressed using $\FF$-limit as $\ol x(\FF)=\Flim x$.$)$
Proof can be found also e.g. in [HS,Chapter 3].
Proof.
$\beta\N$ is Hausdorff. If $\FF\ne\GG$ are two ultrafilters on $\N$, then there is a set $A$ such that $A\in\FF$ and $A\notin\GG$. Then $\UA{A}$ is a basic neighborhood of $\FF$, and $\UA{\N\sm A}$ is a basic neighborhood of $\GG$ and these two neighborhoods are disjoint.
$\beta\N$ is compact. Suppose that there is an open cover $\{\UA{A_i}; i\in I\}$ consisting of basic sets, which does not have a finite subcover. This means that for every finite set $F\subset I$ we have $\beta\N\sm\bigcup\limits_{i\in F}\UA{A_i}\ne\emps$. Since $\beta\N\sm\bigcup\limits_{i\in F}\UA{A_i}=\UA{\N\sm\bigcup\limits_{i\in F}A_i}$, we see that $\N\sm\bigcup\limits_{i\in F}A_i=\bigcap_{i\in F}(\N\sm A_i)\ne\emps$. Hence the system $\{\N\sm A_i; i\in I\}$ has finite intersection property, and therefore there exists an ultrafilter $\FF$ which contains this system. For this ultrafilter $\FF$ we have $\FF\notin\bigcup\limits_{i\in I}A_i$, which contradicts the assumption that $\{\UA{A_i}; i\in I\}$ is a cover of $\beta\N$.
$\beta\N$ is zero-dimensional. All basic set $\UA{A}$ are clopen, since $\UA{\N\sm A}=\beta\N\sm\UA{A}$.
The map $e$ is an embedding. The map $e$ is continuous, since $\N$ has discrete topology. For each $n\in\N$ we have $\Obr e{\{n\}}=\{n^*\}=\UA{\{n\}}$, therefore $e$ is open.
$\Obr e{\N}$ is dense. If $A\ne\emps$, then there is $n\in A$ and we have $n^*\in\UA A$.
Definition of $\ol{x}$. Let $K$ be a compact Hausdorff space. Then for every ultrafilter and any sequence $\Zobr x{\N}K$ there exists a unique limit $\Flim x_n$. For the proof see this question.
Continuity of $\Flim$. Let $\Zobr x{\N}K$, where $K$ is compact. For any $\FF$ we denote $\ol x(\FF)=\Flim x$. Then the map $\ol x$ is continuous. Let $\ol{x}(\FF)=\Flim x=L$. Choose a neighborhood $V$ of $L$ such that $\ol V\subseteq U$. Then the set $A=\Invobr xV$ belongs to $\FF$. We show that $\ol{x}(\GG)\in U$ for any $\GG\in\UA{A}$.
To see this, it suffices to notice that $\Glim x\in\ol{V}$ for each such ultrafilter $\GG$. Indeed, if $\Glim x=L'$ then for any neighborhood $U'$ of $L'$ we have $\Invobr x{U'}\in\GG$. Since $\Invobr xV\in\GG$, we get that $\Invobr x{U'\cap V}=\Invobr x{U'}\cap\Invobr xV\in\GG$. This implies that $U'\cap V\ne\emptyset$. Since every neighborhood of $L'$ intersects $V$, we see that $L'\in\ol{V}$.
Uniqueness of $\ol{x}$ follows from the fact that $\Obr e{\N}$ is dense in $\beta\N$. $\hspace{2cm}\square$
[HS] Hindman, Strauss: Algebra in the Stone-Čech compactification, Walter de Gruyter, Berlin-New York, 1998.
This proof is taken from my notes available here.