there are at most finitely many points

Question

I have the formula $\frac{2a_{i}}{a^{2}_{i}+1}=-b_{i}$ and I want to show that $|a_{i}|=1$ only when $|b_{i}|\geq1$ which is easy to prove but my question is how I can prove the statement " there are at most finitely many $|a_{i}|=1$ ?"

I appreciate any help..

Answer 1

You have a degree 2 polynomial in $a_i$ : b_ia_i^2+2a_i+b_i$ which has a finite number (actually 2) of roots.