Every proper subspace of a normed linear space is either dense or nowhere dense.
Our instructor proved this theorem in our class. I have understood each and every step of the proof. But still I have a confusion. Confusion arose due the following example $:$
Consider the real line $\Bbb R$ with the Euclidean norm. Consider the proper subspace $A=[-1,1] \cap \Bbb Q$ then this is neither dense nor nowhere dense. But if the example is true that would definitely violate the above theorem. What's going wrong here? Inspite of my effort I couldn't find out why is it happening. Please help me in this regard.
Thank you very much.