Norm in a Vector Space A vector space with norm $\parallel\cdot\parallel$ Satisfy for two vectors the following
$\parallel x+y\parallel=\parallel x\parallel +\parallel y\parallel$
i need to proof the fallowing statement
$\parallel \alpha x+\beta y\parallel=\alpha\parallel x\parallel +\beta\parallel y\parallel$
for all $\alpha\geq0$ and $\beta\geq0$, and also this norm it's not necessary induced by an inner product
The fact that 
$\alpha\parallel x\parallel +\beta\parallel y\parallel\geq\parallel \alpha x + \beta y\parallel$
it's clear because the property of the norm, what i can't proof it's the fact that 
$\alpha\parallel x\parallel +\beta\parallel y\parallel\leq\parallel \alpha x + \beta y\parallel$
i will appreciate any help.

I have changed $\alpha, \beta$ to $a,b$. Assume without loss of generality that $b\leq a$. Then $\left\Vert ax+by\right\Vert =\left\Vert
a(x+y)+(b-a)y\right\Vert \geq a\left\Vert x+y\right\Vert -(a-b)\left\Vert
y\right\Vert =a(\left\Vert x\right\Vert +\left\Vert y\right\Vert
)-(a-b)\left\Vert y\right\Vert $
$=a\left\Vert x\right\Vert +b\left\Vert y\right\Vert $ and $\left\Vert
ax+by\right\Vert \leq a\left\Vert x\right\Vert +b\left\Vert y\right\Vert $.
