How do I prove a half open interval is a sigma algebra?

I was reading the lecture notes about probability and random processes, Lecture Notes, then I’m stucked here:

Example 1.3.6 Let $Omega = mathbb R$ and suppose that we have a sigma field A such that all intervals of the form:

$[ 1, 2 – 1/n) in mathcal{A}$

So, I would like to know how to prove that interval is a $sigma$ algebra?

1. $Omega in mathcal{A}$
2. If $A in mathcal{A}$, then $A^{c} in mathcal{A}$
3. If $A_{n} in mathcal{A}$ for each $n$ in a countable collection $(A_{n})_{n=1}^{infty}$, then $cup_{n=1}^{infty} in mathcal{A}$.

I know from the lecture note that:

$[1, 2) in mathcal{A}$, since $lim_{ntoinfty} [1, 2 – 1/n)$

I think its prove statement 3. Am I right?

But I don’t know how to prove statements 1 and 2.