TITLE: Showing that the zero vector has norm zero
QUESTION [5 upvotes]: I need to show that this is a property of a norm.  I know this is supposed to be straightforward but I am somehow not seeing it.
The property is 
$$\lVert 0\rVert = 0$$
I was trying to use the fact that for norms $$ \lVert\lambda \cdot x\rVert = |\lambda|\cdot  \lVert  x\rVert $$ but it seems to be lacking any real reasoning.
EDIT I am using the following for the definition of a norm:
1) $||x|| > 0$ if $x\neq 0$
2) $||\lambda x|| = |\lambda|||x||$ for $\lambda \in \mathbb{R}$
3) $||x+y|| \leq ||x|| + ||y||$
Let me know if you need any more information!

REPLY [7 votes]: By a vector space result, if we pick $\lambda=0$ and $x$ any vector, then $\lambda x$ is the zero-vector.
What you want now follows from (2). With $\lambda=0$, we get the zero-vector on the left. And on the right we get $0$. For on the right we just have multiplication of reals.