mathematical proofs the set of real numbers

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 1

HINT:

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

Answer 2

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 $2n \cdot a > |b|$.