If $p \leq \dim W$ then does $W$ contain linearly independent vectors $v_1, \dotsc, v_p$?

Question

I was trying to understand the solution of a question here on this site and someone gave me a thorough explanation for the solution. But this explanation includes the following fact:

If $W$ is a vector space and $p \leq \dim W$ then there exists a subset $\{v_1, \dotsc, v_p\}$ of $W$ which is linearly independent.

As you can see, the theorem says that for every natural number smaller than the dimension you can find a linearly independent set. I doubt if I have seen this theorem in any book before.

My question is:

Is this fact correct? if yes can you please provide me with a reference for it? if No, could you please give me a counterexample?

Thanks!

Answer 1

There is a basis of size $\dim(W)$ and any subset of size $p$ of that basis is still independent.