Can somebody please check the correctness of this proof, since I am new to this? Thank you in advance.
Given $X$ a normed space and $Y$ a proper subset of $X$ that is a linear subspace, prove that $int Y$ is empty.
Proof: Assume $Y$ is nowhere dense. $X/int(CL (Y)) = CL(X/CL (Y)) = X$ by definition of nowhere dense set. So $Y$ nowhere dense implies $int (CL (Y)) = \emptyset$.
Consequence: The space $X$ belongs to the 1st category of Baire, from this follows a non-positive characteristic of the space, such that: any metric in this space is not complete wrt to topology and the closure of any [non-empty] pen set is not compact.
In the first paragraph of your attempt you "deduce", from $Y$ nowhere dense, that $\text{int}(CL(Y))=\emptyset$. That's usually just the definition of nowhere dense, so as far as I can see you get nothing from the first paragraph.
To do an actual proof, I suggest you try first with $X=\mathbb R^2$ and $Y=\mathbb R$.