Is a subspace of a subspace also a subspace?

[cmh36]

1. True or False: If U is a subspace of V, and V is a subspace of W, U is a subspace of W.

If true give proof of answer, if false, give an example disproving the statement.


2. My thoughts: If U is a subspace of V, then the zero vector is in V. As well as x+v is in V and ax is in V (by definition of a subspace). If these three are in V, and V is in W, then these three must be in W as well. Therefore U will be a subspace of W. However, I don't know if there is an example to disprove this, or if my logic is completely flawed.

Thanks for any help!

 

[radou]

U is a subset of V, and V is a subset of W. This implies that U is a subset of W. Since U is closed under addition and scalar multiplication, we conclude...