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

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?