Let $C_c(\mathbb{R})$ be the set of compactly-supported continuous functions on $\mathbb{R}$. We can view this with a number of different topologies but I have my eye on two in particular. Let $X$ be $C_c(\mathbb{R})$ equipped with the inductive limit topology and let $Y$ be the same set with the compact-open topology.

What is an example of a convergent sequence in $Y$ which fails to converge in $X$?
Intuitively, such a sequence must exist since the map $x\to x$ is not continuous from $Y$ to $X$.