Let $f_m:=\prod \limits_{i=0}^{m}(1+x^{2i+1})$
Show that: $$(f_m)_{m\geq0} \text{ forms a Cauchy sequence in } \mathbb{C}[[]] $$
$\mathbb{C}[[x]]:={\sum_{n\geq0}a_nx^n \text{ | } a_n\in\mathbb{C}}$
I know that a sequence $(x_n)$ is a Cauchy consequence if for all $\epsilon>0$ there is a $N \in \mathbb{N}$ so that $|x_n - x_m| < \epsilon \text{ } \forall n,m \geq N$.
$|\prod \limits_{i=0}^{n}(1+x^{2i+1}) - \prod \limits_{i=0}^{m}(1+x^{2i+1})|=| (1+x)(1+x^3)\cdots (1+x)^{2n+1} - (1+x)(1+x^3)\cdots (1+x^{2m+1}|< \dots ????$
Can you please help me to do that?