Show that if $f_i = O(g), i = 1,...,n$ as $x \to 0$ and..

Question

Show that if $f_i = O(g), i = 1,...,n \space as \space x \to 0 \space and \space |g_i| \leq |g| i = 1,...,n,$ then

$$\sum^n_{i=1}{a_if_i}=O(g), as \space x \to 0,$$

where $a_i, i=1,...,n,$ are constants.

Not really sure how to start this question. Thanks in advance

Answer 1

I suspect $f = O(g)$ means $|f|\leq c(f) g$ where $c(f)$ is a constant depending on $f.$ Then, the exercise follows at once from definition bearing in mind triangle inequality.