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?

Assuming you are dealing with real sequences, notice that for all $n$, one has:
$$0\leqslant {a_n}^2\leqslant{a_n}^2+{b_n}^2.$$
Therefore $(|a_n|^2)_n$ goes to $0$.
Then use the continuity of the square root to conclude.
