I have been reading Richard Silverman "Complex Analysis with applications", I faced a principle which is called principle of nested rectangles I tried to figure out what the textbook is really saying, unfortunately I got confused and didn't understand it completely.
I appreciate it if someone could explain this principle
In addition,I want to know how important this principle is?(Because I haven't seen it in any other Complex Analysis book)
Thanks in advance
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This is simply a generalization of the nested interval theorem mentioned just before it. The idea behind it is that you have a sequence of rectangles in which the sides are always parallel to the real and imaginary axis. If you imagine the length of the diagonals (the diameters) tending to $0$ then the rectangles must be shrinking. We must demand that the diameter tend to $0$ for if we only required it on the volume then it could be that the rectangle shrinks into a line which has $0$ volume but non-zero diameter. Since the rectangles are shrinking in diameter we would expect that there could only be a single point in the limit structure since its diameter is $0$.
This concept seems related to the Cantor intersection theorem (http://en.wikipedia.org/wiki/Cantor's_intersection_theorem).
The author is just introducing the concept because later on the book dives into concepts relating to compactness.
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$\begingroup$ I'm sorry but I didn't clearly understand what you mean by, "We must demand that the diameter tend to 0 for if we only required it on the volume". $\endgroup$– FreeMindAug 3, 2014 at 19:41
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$\begingroup$ If we are interested in having rectangles shrink to a point a natural way to make a sequence of rectangles shrink is to demand that the area decrease to $0$. For one dimensional rectangles this is enough since measure coincides with diameter in this case. In higher dimensions this is no longer true. This is because we can decrease length along a single side which decreases the volume to $0$. For example we can shrink $[0,a]\times[0,b]$ to a line segment of $0$ volume by letting $a$ tend to $0$ but not $b$. $\endgroup$ Aug 3, 2014 at 19:53