Vector norm and relationship with euclidean distance If $y\in E_n$ (n dimensional euclidean space) show that $||\textbf{y}||\leq|\textbf{y}|\leq \sqrt{n}||\textbf{y}||$ 
Where $||\textbf{y}||$ is the euclidean length of the vector $\textbf{y}$
and $|\textbf{y}|$ is the norm of $\textbf{y}$
The hint: Use the inequality $2|uv|\leq |u|^2+|v|^2$ 
to show 
$||\textbf{y}||^2\leq|\textbf{y}|^2\leq n||\textbf{y}||^2$
By letting $u=|y|$ and $v=y$ I get
$2||y|y|\leq||y||^2+|y|^2$ I don't really see how this helps.


Hint: for any nonnegative number $a$ and $b$ you have
$$
(a+b)^2=a^2+b^2+2ab\ge a^2+b^2
$$
and
$$
2(a^2+b^2)-(a+b)^2=a^2+b^2-2ab=(a-b)^2\ge0.
$$
Hence,
$$
a+b\ge\sqrt{a^2+b^2}\quad\text{and}\quad \sqrt2\sqrt{a^2+b^2}\ge a+b.
$$
