Measurable Subset Question

1)Let $E$ be a measurable subset of $\mathbb{R}$ with $m(E)=1$ and let $(E_n)$ be a sequence of measurable subsets of $E$ . If, for any $\epsilon > 0$ , there exists a set $E_i$ in the sequence $(E_n)$ with $m(E_i) > 1 - \epsilon$ , find the value of $m(\cup E_n)$ .

My working is as below:
Since $m(E) = 1$ & $m(E_i) > 1 - \epsilon$ ,
$m(E_i) > m(E) - \epsilon$ , so $m(E \setminus E_i) < \epsilon$ .
So set $\epsilon = \frac{1}{n}$ such that $m(E \setminus E_n) < \epsilon$ for $n \geq i$
So $m(E \setminus \cup E_n) = 0$ .
Hence $m(\cup E_n) = 1$ .

Is this correct?

2)Let $E$ be a measurable set with $m(E) = 1$ . Suppose that ${E_n}$ is a sequence of measurable subsets of $E$ such that $m(E_n) = 1$ for $n=1,2,3,...$ . Find $m(\cap E_n)$ .

I suppose the answer is 1?