Showing that $Y^c$ is dense in $X$, if $ Y \subset (X, \| \cdot \|)$

Question

Exercise :

Let $X$ be a normed space $(X, \| \cdot \|)$ and $Y$ be a proper subspace of $X$, $Y \subset X$. Show that the complement set $Y^c$ is dense in $X$.

Question :

I'm totally at loss on how to start this exercise.

I know that a normed space means it's a linear space carrying the definition and properties of the norm, while on the other hand, a subset $A$ of a topological space $X$ is called dense (in $X$) if every point $x$ in $X$ either belongs to $A$ or is a limit point of $A$; that is, the closure of $A$ is constituting the whole set $X$.

But how would I proceed to showing rigorously the statement of the exercise ?

Answer 1

Let $a\in X$ but $a\notin Y$. Let $y\in Y$. Then $y+ta\in Y^c$ for $t\ne 0$. Now let $t\to0$.