Trouble proving multiplicativity and triangle inequality of a norm.

Question

Let $||*||$ be a norm on a vector space $V$.

$\varphi(v)=\frac{||v||}{1+||v||}$.

I am trying to prove that it's a norm on $V$. I know that I need to prove the 4 properties of the norm. Clearly, $||\varphi||\geq0$. $||\varphi||=0$ iff $\varphi =0$, which is true when $||v||=0$.

But I am having trouble proving that:

1. $||\alpha \varphi || = |\alpha| ||\varphi||$ for $\alpha\in \mathbb{R}$
2. $||x+y||\leq ||x|| + ||y||$ in the context of $\varphi$

EDIT (2): For the triangle inequality,

$\varphi(v+w)=\frac{||v+w||}{1+||v+w||}\leq \frac {||v|| + ||w||} {1+ ||v||+||w||} = \frac {||v||} {1+ ||v||+||w||} + \frac {||w||} {1+ ||v||+||w||} \leq \frac {||v||} {1+ ||v||} + \frac {||w||} {1+ ||w||}= \varphi(v) + \varphi(w)$

Also $\varphi(\alpha v)= \frac {|| \alpha v||} {1+ ||\alpha v||} = \frac {|\alpha| ||v||}{1+ |\alpha| ||v||} \neq |\alpha| \varphi(v)$.