Proving that number is least upper bound

Question

I am to find supremum and infimum of given set, and prove they are in fact supremum and infimum. $$A=\lbrace x\in \Bbb{R}:x^2-5|x|+4\lt0\rbrace $$ $$\sup A = 4, \inf A = -4$$ I tried to prove it in the following way, let $\varepsilon\gt0, a\in A$, then if $a\gt 4-\varepsilon$, $4-\varepsilon$ is no upper bound, so I can set $a$ to be equal to $4-\frac{\varepsilon}{2}$ and therefore $4-\varepsilon$ is not upper bound what proves that 4 is least upper bound. Proof for greatest lower bound is similar. My question is, is my proof correct?

Answer 1

Equivalently, $A = \{x \in \mathbf {R} | 1 < |x| < 4 \}$. This proves that $$\inf A = -4 \qquad \sup A = 4.$$