How to show that a subset of a normed space isn't dense?

Consider an arbitrary normed vector space $(X, \Vert \cdot \Vert_X)$ and let $Y \subset X$. We say that $Y$ is dense in $X$ if and only if

$$ \forall x \in X, \epsilon > 0, \, \exists y \in Y : \Vert y - x \Vert_X < \epsilon. $$

Now, my question is pretty simple:

When we want to prove that $Y$ isn't dense in $X$, what exactly do wehave to show ?

My thoughts. I believe that I was able to provide a condition that answers my question, but I would like to know if there are any weaker conditions that lead me to the same outcome.

So, my idea is as follows. To show that $Y$ isn't dense in $X$, I believe that it is sufficient to find an element $x_0 \in X$ such that there exists a constant $c > 0$ satisfying

$$ \Vert x_0 - y \Vert_X \geqslant c > 0, $$

for every $y \in Y$.

Now, when I mentioned a weaker condition, I was thinking about the following:

If we find an element $x_0 \in X$ such that for every $y \in Y$ there exists a constant $c(y) > 0$ satisfying

$$ \| x_0 - y \|_X \geqslant c(y) > 0, $$

does it also follow that $Y$ isn't dense in $X$ ?

Thanks for any help in advance.