Is this the way to prove it?

Question

I want to prove that

dim($W$) $\le$ dim($V$) if $W$ $\subseteq$ $V$

So I did it this way

If $W$ $\subseteq$ $V$ then the maximum number of linearly independent vectors in $W$ $\le$ the maximum number of linearly independent vectors in $V$

Since the dimension of a vector space is the maximum number of linearly independent vectors in it we can say

dim($W$) $\le$ dim($V$) if $W$ $\subseteq$ $V$

Is this the right method to do this?

Answer 1

Yes.

A basis of $W$ is an independent subset of $V$ which can be further extended to form a basis of $V$.

Hence $\operatorname{dim}(W) \le \operatorname{dim}(V)$.