Let ${a_n}$ and ${b_n}$ be sequences. Prove that if $\lim_{n\rightarrow \infty}(a_n^2 + b_n^2) = 0$ then $\lim_{n\rightarrow \infty}a_n=0$ Let ${a_n}$ and ${b_n}$ be sequences. Prove that if $\lim_{n\rightarrow \infty}(a_n^2 + b_n^2) = 0$ then $\lim_{n\rightarrow \infty}a_n=0$ 
This was a question on my Real Analysis undergrad exam. Which I tried to prove by wrongly using the contrapositive. Can someone direct me on the right path?

Hint Let $\epsilon >0$. If $N$ is such that ${a_n}^2+{b_n}^2 < \epsilon^2$ for all $n >N$, show that 
$$\left| a_n -0 \right| < \epsilon \qquad \forall n >N$$
