I am trying to solve the following problem:
Consider the product space $(\mathbb{R}^\mathbb{N},\bigotimes_{n\in\mathbb{N}}\mathfrak{B})$ with $\mathfrak{B}$ the Borel $\sigma$-algebra on $\mathbb{R}$. Show that $\bigotimes _{n\in\mathbb{N}}\mathfrak{B}$ contains the sets of all bounded, monotone and convergent real sequences.
I could show that $\bigotimes _{n\in\mathbb{N}}\mathfrak{B}$ contains the set of all bounded sequences by considering the sets $\{(a_n)_{n\in\mathbb{N}}:-\alpha<a_n<\alpha\ \forall n\in\mathbb{N}\}$ for all $\alpha\in\mathbb{N}$.
For monotone and convergent sequences however I can only think of uncountable ways to define the sets. Can someone give me a hint on how to solve the rest?
Thanks in advance!