If is a Hamel basis for a vector space with elements, then every subset of having elements are not linearly independent.

Proof:

I am doing this by inductions. I've shown it's true when .

Assume it's true when .

I have trouble with the case when .

Let be a Hamel basis, and be a subset of X. Let be the collection of all linear combinations of .

How do I show that B is not linearly independent?