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.

If $n \rightarrow \infty$ then $O(\frac{1}{n}) \rightarrow 0$ and using $$\frac{1}{1-z} = \sum_{n=0}^{\infty}z^{n} \ \ \ \ , \ \ |z|<1$$ we have $$\frac{1}{1 + O(n^{-1})} = \sum_{k=0}^{\infty}(-1)^{k}O(\frac{1}{n})^{k} = 1 +O(\frac{1}{n})$$
