I think this is quite a basic question but I'm struggling with it. Suppose $x_1, x_2, \ldots x_n$ are a set of finite-dimensional vectors and $a = (a_1, a_2, \ldots, a_n)$ and $b = (b_1, b_2, \ldots, b_n)$ are $n$-dimensional vectors of scalars. In the solution to a problem, we have:
\begin{align}\max_i ||x_i|| \cdot \sum_{i=1}^n |a_i - b_i| &= \max_i ||x_i|| \cdot ||a-b||_1\\& \leq \sqrt{n}\max_i ||x_i|| \cdot ||a-b||_2 \end{align}
Could someone please explain the last inequality? I understand that $||a-b||_2 = \sqrt{\sum_{i=1}^n |a_i-b_i|^2}$ and so $\sqrt{n} ||a-b||_2 = \sqrt{n\sum_{i=1}^n |a_i-b_i|^2}$ but wasn't sure how to show that $\sqrt{n\sum_{i=1}^n |a_i-b_i|^2} \geq \sum_{i=1}^n |a_i - b_i|$.
