I am wondering what it means to prove something via the saturation argument. I have seen this appear in a number of questions, but I am not sure as to what a saturation argument is. Any help would be appreciated.
Original Post: Saturation Argument in Analysis: Everything is in the title. I'm French and I don't understand the meaning.
Examples of where this has appeared:
Example 1: By a saturation argument, show that $F$ is continuous on $(-1,1)$ where $F$ is defined as $$F(x) := \int_{-\pi }^{\pi }\ln (1 + x^2 - 2x\cos \theta) d\theta, \quad x \in (-1,1).$$
Example 2: By a saturation argument, show that the series $S$, defined as $$S(x)=\sum_{n\geq 1}\frac{1}{n^x}$$ is continuous on $(1,+\infty)$.
Places where this phrasing appears includes: http://www.math.hawaii.edu/~ross/Math699/lecnotes.pdf