$\frac{a_{max}}{\sum a_i} \to 0$: Proof or Counterexample

Suppose I have a sequence of positive integers $\{a_n\}$. Let us denote $b_n=\max_{1\le i\le n} a_i$. Suppose $$\frac{b_n}{\sum\limits_{i=1}^n a_i} \to 0$$ then show that $$\frac{b_n^2}{\sum\limits_{i=1}^n a_i^2} \to 0$$

I am not sure if it is true. But I didn't find any Counterexample. I was trying to get a reasonable lower bound for the denominator. I could not find any. Bounds like $$\sum_{i=1}^n a_i^2 \ge \sum_{i=1}^n a_i$$ won't help though. Note that the converse is true. As you can easily get an upper bound using: $$\sum_{i=1} a_i^2 \le b_n\sum_{i=1} a_i$$

Any help/suggestions?

Edit: Note that $a_n$'s are positive integers, that's why $\sum a_i^2 \ge \sum a_i$ is true.

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