Question about Hölder's inequality proof

Question

Let $1 < p,q < \infty$ with $\frac{1}{p} + \frac{1}{q}=1$. Then, for $a,b \in \mathbb{K}^n$, we have:

$$|\sum_i a_ib_i| \leq \Vert a \Vert_p \Vert b \Vert_q$$

My book gives the proof in the assumption that $a_i,b_i \geq 0$. Why can we make this assumption?

Answer 1

Because then \begin{align*} \left|\sum a_{i}b_{i}\right|&\leq\sum|a_{i}||b_{i}|\\ &\leq\|(|a_{i}|)\|_{p}\|(|b_{i}|)\|_{q}\\ &=\|(a_{i})\|_{p}\|(b_{i})\|_{q}. \end{align*}