If $bb_n \rightarrow b^2$ how do I show $bb_n>\frac{b^2}{2}$

Question

This is the only part I'm struggling to show in my proof of the quotient rule for sequences. I've tried setting $\epsilon$ to some values but I don't think what I am doing is correct.

Answer 1

HINT

If $bb_n \rightarrow b^2$ we have that $\forall \epsilon \quad \exists \bar n$ such that

$$\forall n\ge \bar n \quad |bb_n-b^2|<\epsilon \iff b^2 -\epsilon < bb_n < b^2 + \epsilon$$

then choos a suitable $\epsilon$.