Prove that the set consists of (un)bounded sequences

Question

I don't understand the highlighted statement, how can one prove this?

Answer 1

If $a$ is bounded then $|a_i|<M$ for all $i$ therefore every point in $(a_{i-1},a_{i+1})$ is bounded by $M+1$

Thus every sequences in the Cartesian product is bounded.

Similarly you can argue for the unbounded case .