# A map to a larger dimensional space is not surjective

I'm reading Axler's Linear Algebra Done Right book and its my first time doing linear algebra. I'm confused on this proof: Suppose V and W are finite-dimensional vector spaces such that dim V < dim W. Then no linear map from V to W is surjective. Proof: Let T ∈ L(V, W).
Then dim range T = dim V - dim null T ≤ dim V < dim W I understand the first equality (fundamental thm of linear maps). the last inequality i understand comes from the assumption. I'm confused on the middle inequality. Any help would be appreciated.

The dimension of the null space is weakly greater than $$0$$. $$a - k \leq a$$ if $$k \geq 0$$.
Generally speaking, if $$x$$ and $$y$$ are numbers and $$y$$ is nonnegative, meaning $$y\geq0$$, then $$x-y\leq x$$.
In this case, $$\dim\mathrm{null}\;T\geq 0$$, so $$\dim V - \dim\mathrm{null}\;T \leq \dim V$$.