## Prove that if a_{n} \to Aa_{n} \to A for some real AA then a_{n}^{2} \to A^{2}a_{n}^{2} \to A^{2}

Let A \in \mathbb{R} and let (a_{n}) be a sequence of reals convergent to A. We have |a_{n}^{2} – A^{2}| = |a_{n} – A||a_{n} + A|. I am stuck at majorizing |a_{n} + A|. I know that convergent sequences are bounded, but, that kind of bounds, which takes the form \max \{ |a_{1}|, \dots, |a_{N-1}|, … Read more