Assume that $a_j\leq u$ and $p_j\in\{-1,+1\}$ for $j=1,2,\ldots,n$ and that $nu<1$ where $u:=2^{-t-1}$. Show that following is true: $$\prod^{n}_{j=1}(1+a_j)^{p_j}=1+\theta_n$$ where $|\theta(n)|\leq\gamma_n$ for which it is true that $\gamma_n:=\frac{nu}{1-nu}$
This is question from here (p. 63), but unfortunately it's given without a proof.
I tried to split that lemma into two parts, where all $a_k$ have their $p_k$ with value of 1 and all $a_l$ have $p_l$ of -1, but I don't see what's next.