This is the first part of another section in the 'Introduction to real analysis' series
When working with sequences and series, especially in order to prove convergence or divergence of sequences and series, several 'tests' are often employed.
These tests (especially in the case of positive series) usually end up making use of the 'least upper bound' or 'completeness' property of the Real Numbers as a core argument. (Although this is usually well disguised, as comparison tests make no reference to this property - although this property is vital in the actual proof and therefore legitimacy of said comparison tests)
So, before we start working with sequences or series of Real Numbers, I thought it would make sense to present a construction of the Real Numbers which will first establish the completeness or 'least upper bound' property of the Real Numbers so there isn't any ambiguity or confusion when I reference it in any of the proofs regarding tests of convergence that will follow.
There are two conventional ways of constructing the Real Numbers, one approach is to define Real Numbers as sets called 'Dedekind cuts' (which are intuitively partitions of the number line), while the other approach involves defining Real Numbers as equivalence classes of so called 'Cauchy sequences'.
This second approach essentially involves saying that as long as you have a sequence of rational numbers, where terms can be made as close to each other as you wish as long as you take terms far enough into the sequence, (so we can make $|u_n - u_m| < \epsilon$ as long as both $n$ and $m$ are larger than some natural number $N$), then this sequence 'ought to' converge. Unfortunately not all of these sequences will converge to a number in the rationals, and so in order to fix this problem we define the set of all real numbers as the 'completion' of the rationals in the sense that all sequences of this type always converge to some 'real number'.
Another way of going about better visualising how we might construct the Real Numbers is to imagine an infinitely long strip of paper, and a machine that repeatedly lowers and raises a pen upon this paper while moving horizontally (back and forth) along the strip. The machine (as it moves horizontally back and forth along the paper) continues to lower and raise the pen. If the machine has moved horizontally back and forth between some spot and another spot, all of its oscillatory movements after this will be contained within this interval. Furthermore, the machine must eventually oscillate with a smaller oscillatory width after a period of time, and will continue to oscillate back and forth with ever decreasing oscillatory width. Moreover, its new oscillations will always be contained within its previous oscillations. After a long period of time, we would be able to say with confidence that the machine is 'trapped' within a small interval. Over time this interval would get smaller and smaller, and would narrow in on a 'point' on the piece of paper, this would be a 'Real Number'.
Both Cauchy and Dedekind's constructions require prior constructions of the set of all rational numbers.
The construction of the Real Numbers that I will present here will be similar to the construction via Cauchy sequences, but will perhaps support the 'machine - pen' description a bit more vividly.
(I will also be assuming a prior working construction of the rationals when presenting my construction of the Real Numbers).
What this means is that we will be assuming that the set of rational numbers has already been constructed, and is equipped with all the nice properties we expect the rational numbers to be equipped with.
Now, before we can get started lets start with some intuition to support the construction I will present:
Imagine the 'Real Number' line was in front of you. Now mark a point on the line, and another point to the right of this point with a left and right closed bracket respectively. Continue to do this, but make sure that all new left closed brackets you mark are to the right of previously marked left closed brackets, and that all right closed brackets you mark are to the left of previously marked right closed brackets, and that there are no left closed brackets to the right of right closed brackets. What you should begin to notice after a while, is that both closed brackets begin to 'close in' to something.
This can actually be summed up in a theorem, which will be stated and proved in the next part of the 'Sequences, Series and the Real Numbers' section. Furthermore, this theorem will be the key to our construction of the Real Numbers which will also follow.
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