# Prove $\sum \sqrt{a_n b_n}$ converges if $\sum a_n$ and $\sum b_n$ converge.

Prove $\sum \sqrt{a_n b_n}$ converges if $\sum a_n$ and $\sum b_n$ converge.

I can prove that $\sum a_n b_n$ converges but couldn't for $\sum \sqrt{a_n b_n}$.

Thank you.

EDIT: $a_n, b_n \ge 0 \;\forall n$

• I assume that $a_n, b_n\ge 0$... ? – mookid Oct 27 '14 at 20:40
• The question doesn’t state that $a_n,b_n\ge 0$. Without this it is clearly false by taking both to be the same conditionally-convergent series. – Erick Wong Mar 12 '18 at 17:25

Hint: $$\sqrt {AB}\le A+B$$ because of $(\sqrt A - \sqrt B)^2 \ge 0$.
Note: Similar can be shown to hold true for $A_1, ..., A_n$ and $A_1, A_2, ...$
• We assume that it can be shown by induction that similar holds true for both $A_1,A_2,...,A_n$ and $A_1, A_2,...$ ? – BCLC Mar 8 '18 at 16:16
• mookid, the point is the extension to $\infty$ is true but not trivial... – BCLC Mar 12 '18 at 17:17