Understanding $ \int_c^\infty (x-c) dF(x) $ through integration by parts

Question

Following this answer, it is claimed that we can solve the problem in the following way:

$$\displaystyle \int_{c}^{\infty} (x-c) dF(x) = \lim_{y \rightarrow \infty} (y-c) F(y) - \displaystyle \int_{c}^{\infty} F(x) dx.$$

where $F$ is the cumulative distribution function of a given random variable.

Does this limit exist though? In my understanding, since distribution function saturates at 1, the first limit should converge to infinity? What am I missing?

Answer 1

Shouldn't the formula be,

$\displaystyle \int_c^\infty dP(x) = \lim_{y\to\infty} \left\{ (y-c) P(y) - \int_c^y P(x) dx \right\} $

Then for large $ x $, $ P(x) $ approaches 1, and you end up with both terms approaching $\infty$.

Not very helpful in finding the answer but perhaps explains the confusion.