Proof of square root properties

[diracy]

Homework Statement

\sqrt{\sum}x^{2}_{i}\leq\sum|x_{i}|\leq\sqrt{n}\sqrt{\sum}x^{2}_{i}

*The sums are all from i=1 to n.*

Homework Equations

The Attempt at a Solution

I'm very new to proof-based math, and just looking for some help to get started with this one. Thanks in advance.

 

[lanedance]

For this part
\sqrt{\Sum_i x_i^2} \leq \Sum_i |x_i|

it should be clear
0 \leq \sqrt{\Sum_i x_i^2}
0 \leq \leq \Sum_i |x_i|

so squaring both sides could be useful

 

[diracy]

I tried that and didn't get far. It seems to my the leftmost inequality is always equal. I must be thinking about it wrong. In what instance would that inequality be less than?

 

[lanedance]

you should some cross terms like |xi||xj| on in the middle, which don't appear on the left