Consider a norm on Rn.
there are two vectors x,y in Rn
x=(x1,x2,x3,...,xn)
y=(y1,y2,...,yn)
if, y1>x1,
y2>x2,
y3>x3,
.
.
.
yn>xn
.
is it true that norm : ||y||>= ||x|| for any norm on Rn?
yes, i know what the norm is defined as. i asked you to compare $\displaystyle \| y \|^2$ with $\displaystyle \| x \|^2$ (we can do this since we were given comparisons component wise).
anyway, just make a counter example, it shouldn't be hard. consider vectors in $\displaystyle \mathbb{R}^2$ given by
$\displaystyle \vec y = \left< 1,1 \right>$ and $\displaystyle \vec x = \left< -5, -5 \right>$
do those vectors satisfy the conditions given? is $\displaystyle \| y \| \ge \| x \|$ ?
$\displaystyle \begin{array}{ccl} \|y \|^2 & = & \left( \sqrt{y_1^2 + y_2^2 + \cdots + y_n^2} \right)^2 \\
& & \\
& = & y_1^2 + y_2^2 + \cdots + y_n^2 \\
& & \\
& \ge & x_1^2 + x_2^2 + \cdots + x_n^2 \\
& & \\
& = & \| x \|^2 \end{array}$
the result follows by taking the square roots of both sides