Could you tell me how to prove such lemmas?
We are given a decreasing sequence of positive numbers $(a_n)$
1) If $\lim_{n \rightarrow \infty} \frac{a_{2n}}{a_n} =g< \frac{1}{2}$ then the series $\sum _{n=1} ^{\infty} a_n$ is convergent
2)If $\lim_{n \rightarrow \infty} \frac{a_{2n}}{a_n} =G > \frac{1}{2}$ then the series $\sum _{n=1} ^{\infty} a_n$ is divergent
I know it must be very easy but I don't know what to do about $2n$ in $a_{2n}$ (which I gather is the main issue of the whole proof).
Do you think you could help me?
Thank you.