Suppose $X_n$ is a sequence of random variables on $(\O,\F,\P)$. $X_n$ converges $\P$-a.s.f.s. if and only if
$$
\forall\e > 0:\quad
\lim_n\P\Big(\sup_{k\geq n}|X_k-X_n| > \e\Big)=0~.
$$
$X_n$ converges a.s, iff $X_n$ is a.s Cauchy. This is equivalent to
\begin{eqnarray*}
0
&=&\P\Big(\limsup_{j,k}[|X_j-X_k| > \e]\Big)
=\P\Big(\bigcap_n\bigcup_{j,k\geq n}[|X_j-X_k| > \e]\Big)\\
&=&\lim_n\P\Big(\bigcup_{j,k\geq n}[|X_j-X_k| > \e]\Big)
=\lim_n\P\Big(\sup_{j,k\geq n}|X_j-X_k| > \e\Big)
\end{eqnarray*}
Now we have by the triangle inequality
\begin{eqnarray*}
\P\Big(\sup_{j,k\geq n}|X_j-X_k| > \e\Big)
&=&\P\Big(\sup_{j,k\geq n}|X_j-X_n-X_k+X_n| > \e\Big)\\
&\leq&\P\Big(\sup_{j\geq n}|X_j-X_n|+\sup_{k\geq n}|X_k-X_n| > \e\Big)\\
&\leq&\P\Big(\sup_{j\geq n}|X_j-X_n| > \e/2\Big)
+\P\Big(\sup_{k\geq n}|X_k-X_n| > \e/2\Big)~.
\end{eqnarray*}