Prove $\sum_{n=1}^\infty(\frac{1}{a_{2n-1}}-\frac{1}{a_{2n}})$ convergent Let $(a_n)_{n=1}^\infty$  Let be a positive, increasing, and unbounded sequence. Prove that the series:
$$\sum_{n=1}^\infty\left(\frac{1}{a_{2n-1}}-\frac{1}{a_{2n}}\right)$$
convergent.

We know that since $a_n$ is increasing and unbounded, than $\lim_{n \to \infty}a_n=\infty$, so I want to apply that to say that, $\sum_{n=1}^\infty(\frac{1}{a_{2n-1}}-\frac{1}{a_{2n}})$ is decreasing and its limit will be $0$.
My problem is that every time I get confused while it says $a_{2n}$ or $a_{2n-1}$, it is less intuitive for me than just "normal" $a_n$...
Appreciate your help!
Thanks a lot!

Hint: Let
$$b_n=\frac1{a_{n-1}}-\frac1{a_{n}}$$
Each $b_n$ is non-negative, since $a_n$ is increasing.
Then your series is: $$\sum_{n=1}^\infty b_{2n}$$ But:
$$\sum_{n=2}^{2N} b_n\geq  \sum_{n=1}^{N} b_{2n}\geq 0$$
And $$\sum_{n=2}^{2N} b_n=\frac1{a_{1}}-\frac1{a_{2N}}.$$
