Is it valid to extend Cauchy-Schwarz to the 3 variable case with the following proof:
$$ \sum_{i=1}^n(a_ib_i) c_i \leq \sqrt{(\sum_{i=1}^{n}a_i^2b_i^2)(\sum_{i=1}^{n}c_i^2) } \leq \sqrt{\sum_{i=1}^{n}a_i^2} \sqrt{\sum_{i=1}^{n}b_i^2} \sqrt{\sum_{i=1}^{n}c_i^2} $$
The first inequality simply applies Cauchy-Schwarz to the two variable case. The last inequality assumes $\sum_{i=1}^{n}a_i^2b_i^2\leq (\sum_{i=1}^{n}a_i^2)(\sum_{i=1}^{n}b_i^2)$. Is this a valid assumption?
Yes, and your proof works as you suggest. You can complete it by expanding the product $(\sum_{i=1}^n a_i^2)(\sum_{i=1}^n b_i^2)$ and observing that the terms are all nonnegative and include all the terms of $\sum_{i=1}^n a_i^2b_i^2$.