Let $R^\infty$ be the vector space of all sequence $\{a_j\}$ of real numbers. Put $\|\{a_j\}\|_n:= \sum_{j=0}^n |a_j|$. This collection of semi norms make this as Frechet space.
A set $B$ is bounded if every continuous seminorm is bounded on $B$. That is bounded set will be as $\{a=\{a_i\}: \sum_{i=0}^k |a_i|<M_k\text{ for some } M_k>0 \text{ and } \forall k\in \mathbb N\}.$
Can I have an explicit example of one bounded set and one unbounded set.
On
The set $\{(x_n)_{n\in N}\}$ with the sequence $x_n=0$ is bounded. Define the set $B=\{(a_n^1),(a_n^2),...,(a_n^m),...\}$ such that $a_n^m = n$, if $n\leq m$ and 0 otherwise. It is clear that $|(a_n^m)| = \sum\limits_{i=1}^{m} i = \frac{(m+1)m}{2}$. Since the value of the norms are unbounded the set is unbounded.
We can check that a subset $B$ of $\Bbb R^{\infty}$ is bounded if and only if all the subsets of $\Bbb R$ defined by $$B_k:=\{x_k,(x_n)_{n\in\Bbb N}\in B\}$$ are bounded.
For example, the set $\{(x_n)_{n\geq 1}\},|x_n|\leq n\}$ is bounded.