Let ( x n ) n ∈ N {\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }} be a null sequence and let ( y n ) n ∈ N {\displaystyle {}{\left(y_{n}\right)}_{n\in \mathbb {N} }} be a bounded real sequence. Show that also the product sequence ( x n y n ) n ∈ N {\displaystyle {}(x_{n}y_{n})_{n\in \mathbb {N} }} is a null sequence.
