Prove Lower bound of a set ${x^3 : x<10}$

Question

I just started studying Calculus and I'm confused on how to prove Lower bounds. I understood how to prove upper bounds but I just don't know how to proceed.

let $B={({x^3 : x<10 , x∈R})}$

The upper bound is obviously $m=1000$ and all the numbers that are bigger than that.

Lower bound - I assumed that exists lower bound $y∈R$ such that every $b∈B$ : $b≥ y$.

and Then I'm stuck , I don't know how to proceed with the proof to show that no such $y$ exists.

Answer 1

Let $b$ be the lower bound (certainly a negative number), i.e. $\forall x<10:x^3\ge b$.

But if we take $x=2\sqrt[3]b$, we do have $x<10$, but $x^3=8b<b$, a contradiction.