Question

Prove that 3Z is infinite

Answer 1

We known that $\Z = \{ ... , - 3,-2,-1,0,1,2,3, ... \}$ .

Hence, $3\Z = 3 \times\{ ... , - 3,-2,-1,0,1,2,3, ... \} = \{ ... , - 9,-6,-3,0,3,6,9, ... \}$ .

By completeness property of real numbers, for every $k \in \R$ , $\exist \ n \in \Z : n \geq k$ . Hence, $\Z$ is an infinite set.

Since, $\Z$ is an infinite set, so scalar multiple of $\Z$ is also infinite set. Hence, $3\Z$ is an infinite set.