Borel σ-algebra generated by intervals on a real line

CA watch $ B _ { mathbb R} = σ (A) $, when A is:

(a) {$ (a, b): a, b ∈ R $}

(B) {$ (- ∞, a): a ∈ R $}

(B) {$ (-, a]: a R $}

(re) {$ (a, ∞): a ∈ R $}.

The proof must not be strict.

My solution:

For (a) I discovered that the interval $ (a, b) $ can be written as:

$ (a, b) = bigcup_ {n = 1} ^ infty (a, b- frac {1} {n}) $ or $ (a, b) = bigcap_ {n = 1} ^ infty (a, b- frac {1} {n}). $

But, however, I do not know how to show that $ B _ { mathbb R} = σ (${(a, b): a, b ∈ R}$) $.