What are some interesting topological properties about the set of all convergent real sequences? Let C be the set of all convergent sequences in $\mathbb{R}$ under the product topology or uniform topology. So $C \subset \prod_{n=1}^\infty \mathbb{R}$. For one, Hilbert's Cube $\prod_{n=1}^\infty \left[0,\frac{1}{n}\right]$ is contained in $C$. 

Let $\ \lim:C\to\mathbb R\ $ be the limit function (functional). Then
$$ \lim(G)\ =\ \mathbb R $$
for every non-empty open subset of the product space. It follows that the limit function is not continuous under the product topology.
