Question: Let $a,b \in \Bbb{R}$. Suppose that $a\gt 0$. Prove that there exists some $n \in \Bbb{N}$ such that $b \in [-na,na]$

Answer: 
**Hint:** Show that 

1) The set of $\mathbb N \subset \mathbb R$ is not bounded from above. 

2) Use (1) to show that given $a,b \in \mathbb R^+$ there exists $n \in \mathbb N$ such that $n \cdot a > b$. 

Conclude that there exists $n \in \mathbb N$ such that $2n \cdot a > |b|$.
**HINT**:

Consider the number $k = \left \lceil \frac {|b|}{a} \right \rceil $