Let $(a_n)$ be a sequence of positive real numbers, Such that $(a_1+a_3+a_5+\dots +a_{2n-1})(\frac{1}{a_1}+\frac{1}{a_2}+\dots+\frac{1}{a_{2n-1}})\geq(2n-1)^2$.Then which of the following is/are true?
1.$(a_n)$ is a Cauchy Sequence.
2.$(a_n)$ can be a cauchy sequence.
3.$(a_n)$ is a monotonic sequence.
4.$(a_n)$ has a convergent subsequence.
I tried to construct some counters to discard the options but eventually most of them were not satisfying the given condition. For instance in order to discard (3) we can have $(a_n)=(2,3,2,3,\dots) $(but this is not correct!). A sequence of real numbers is convergent iff its cauchy and to have a convergent subsequence $(a_n)$ should be bounded too.How can I use the given condition to get some conclusion out? Any help? Thanks.
Too long for a comment:
To show $2)$ is true, consider $a_n=\frac{1}{n^3}$. Then according to Mathematica
$$\lim_{n\to\infty}\frac{\left[\sum_{i=1}^{n}a_{2i-1}\right]\left[\sum_{i=1}^{2n-1}\frac{1}{a_i}\right]}{(2n-1)^2}=\infty$$
and holds true for all $n$.
To show $4)$ is false, consider the sequence
$$a_n=\begin{cases} n & \text{if } n\text{ is even}\\ n^2 & \text{if } n\text{ is odd} \end{cases}$$
Then again, according to Mathematica
$$\lim_{n\to\infty}\frac{\left[\sum_{i=1}^{n}a_{2i-1}\right]\left[\sum_{i=1}^{2n-1}\frac{1}{a_i}\right]}{(2n-1)^2}=\infty$$