Prove that a subset $W$ of a vector space $V$ is a linear subspace of $V$ From Friedberg, 4th edition:


  Prove that a subset $W$ of a vector space $V$ is a subspace of $V$ if
  and only if $W \neq \emptyset$, and, whenever 
   $a \in F$  and $x,y \in
 W$, then $ax \in W$ and $x + y \in W$.


I understand that in order to be a subspace, $W$ must contain the element $0$ such that for any element $x \in W$, it satisfies $x+0=x$. If $W$  is null, it cannot be a subspace, therefore its contrapositive statement must be true: If $W$ is a subspace it cannot be null.

I also understand that a subspace must be closed under addition and scalar multiplication. $W$ follows both these properties as stated in the question, but I am unsure as to how to 'prove' this. Aren't the conditions set in the question the definition of what it means to be a subspace? Is it sufficient to state that $W$ satisfies both these conditions and is therefore a subspace?
From the question, we know that $W$ is closed under addition and scalar multiplication. So all we have to do to prove that $W$ is a subspace is to show that $W$ contains the zero vector. 
So, if $x$ is in $W$, and $ax$ is in $W$, then we can take $a=-1$. By this, we have -$x$ is in $W$ (Closure under scalar multiplication). Now, because we know that $W$ is closed under addition, we have $x+(-x)=x-x=0$, which is in $W$, thus showing that $W$ must contain the zero vector.
Is it okay to use 0,-1\in K as we never ask the existence of 0 or -1\in K in the definition of vector space