Equation with the big O notation How I can prove equality below?
$$
\frac{1}{1 + O(n^{-1})} = 1 + O({n^{-1}}),
$$
where $n \in \mathbb{N}$ and we are considering situation when $n \to \infty$.
It is clearly that it is true. But I don't know which property I have to use to prove it formally. I will appreciate if someone could give me some clues.

Assume $f\in O(n^{-1})$. Then there exists $c\in\mathbb R$ and $n_0\in\mathbb N$ such that  $|f(n)|<\frac cn$ for all $n>n_0$. Wlog. $n_0>2c$. Then for $n>n_0$ we have $1+f(n)>1-\frac cn>1-\frac c{n_0}>\frac12$ and hence 
$$ 0<\frac1{1+f(n)}<\frac1{1-\frac cn}=1-\frac{c}{n-c}$$
and  from $-\frac c{n-c}\in O(n^{-1})$ the claim.
