Here is an explanation of selberg's proof of prime number theorem.
Here by using mertens theorem they show that,
$$\sum_{k≤n}\frac{R(k)}{k^2}=O(1)$$
And then proceed to show that,
$$\left|\frac{R(y)}{y}\right|<\frac{K_{2}}{\log(x'/x)}$$
The explanation to this is shown in the picture.
My question:-
$(1)$ Why does $$\sum_{x≤n≤x'}\frac{1}{n}\inf_{x≤y≤x'}|\frac{R(y)}{y}|≤|\sum_{x≤n≤x'}\frac{R(n)}{n^{2}}|$$
There seems to be a mistake there (no judgment on whether it's a major or minor mistake in context).
For every $n\in[x,x']$, we certainly have $$ \inf_{x\le y\le x'} \biggl| \frac{R(y)}y \biggr| \le \biggl| \frac{R(n)}n \biggr| $$ by the definition of infemum. Applying this to each term yields $$ \sum_{x\le n\le x'} \frac 1n \inf_{x\le y\le x'} \biggl| \frac{R(y)}y \biggr| \le \sum_{x\le n\le x'} \biggl| \frac{R(n)}{n^2} \biggr|. $$ However, this does not immediately imply $$ \sum_{x\le n\le x'} \frac 1n \inf_{x\le y\le x'} \biggl| \frac{R(y)}y \biggr| \le \biggl| \sum_{x\le n\le x'} \frac{R(n)}{n^2} \biggr| $$ as claimed. Certainly for abstract functions $R(t)$, it's possible for the right-hand side to equal $0$, while the left-hand side is always positive when $R(t)$ is not identically $0$.