Why is $\operatorname{cov}(X_i,X_j) \leq 1/\min(i,j)$?

Question

I'm reading this proof and I understand everything except the step marked by an arrow where $\operatorname{cov}(x_i,x_j)< 1/\min(x_i,x_j)$.

I do not understand how they got that step directly and the consecutive step of getting that summation after.

Answer 1

Check that $\max\big(\frac{1}{a},\frac{1}{b})= \frac{1}{\min(a,b)}$

where $a$ and $b$ are positive numbers