Real numbers as solutions to infinite systems of equalities
In the past we already extended our number system by adding “missing” elements which are assumed to satisfy certain equations, based only on knowing what these equations are. It turns out that we may extend the set of rational numbers 
On one leg, the idea can be explained as follows. Since for any two rational numbers 
However, sometimes the analogous construction leads to problems. For instance, if 
Proceeding this way, we introduce (positive) real numbers by indication, what is their relative position to all rational numbers. This allows to describe the real numbers completely.
The details can be found here.
A didactic digression
Some of you complained about insufficient number of problems that are discussed during the tutorials. Everybody knows that problems and questions for self-control are the most important elements of study mathematics, especially in comparison with other disciplines. The rationale behind is the assumption that a student who understands the subject, should be able to answer these questions immediately or after some reflection. Composing such problems is an easy thing: you any mathematical argument you can stop for a second and ask yourself: “why I can do as explained?” or “under what conditions are my actions justified?”. In the lecture notes (see the link above) tens of such problems are explicitly formulated. Similar problems will await you on the exam.
However, remember one simple thing. If you already know how to solve a problem, this is not a problem but rather a job. Unless you solve these problems yourselves, there is no sense in memorizing their solutions: knowing solution of one such problem won’t help you with solving another problem unless you really understand what’s going on. There are no “typical problems”: each one of them is of its own sort, though, of course, some problems can be solved by similar methods.
A practical advice: you should not expect that all problems that appear on the exam will be discussed at length at the tutorials. There are no ready recipes to memorize. Only to understand honestly. Believe me, this is easier than memorize by heart endless formulas and algorithms.
is negative at 
, positive at 
but does not take the intermediate value zero: 
. Another situation concerns the possibility to define the notions of supremum and infimum for infinite sets: the set 
is bounded from two sides, but among its upper bounds 
there is no minimal one.
one can associate two subsets 
as follows: 
and 
. Then the number 
is the unique solution to the infinite system of inequalities of the form 
for different choices of 
. This system has the following two features:
is no greater than any upper bound 
, and
, and none of the sets can be enlarged without violating the first condition.
satisfying the two conditions above.
for all 
as above, and this happens if and only if 
. Yet not all cuts have rational solutions (give an example!).
be two different cuts. We declare that 
, if 
, i.e., if there is a rational number 
and a lower bound for the cut 
. If both cuts have rational solutions, this number would be squeezed between these solutions. In the similar way we define the opposite order 
if and only if 
.
one and only one of the three possibilities holds: 
or 
(meaning that 
). This is a routine check: if the first two possibilities are excluded, then 
, and therefore 
is a self-consistent cut. But because of the maximality condition, this means that 
and 
, that is, 
, then these solutions satisfy inequalities 
(check that you understand the meaning of this inequality between sets and numbers ;-)!) Adding these inequalities together means that 
satisfies the infinite system of inequalities 
, where 
stands for the so called Minkowski sum 
(the same for 
). This allows to define the summation on cuts.
is the cut 
with the Minkowski sum in the right hand side.
as follows, 
, where (of course!) 
. Note that the upper and lower bounds exchanged their roles, since multiplication by 
changes the direction (sense) of the inequalities. Then we can safely define 
as 
. Again, one has to check that this definition is well-behaving and all arithmetic properties are preserved.
(do it yourselves!) and then extend it for negative cuts and the zero cut. After introducing this definition, one has to make a lot of trivial checks:
which admits at least one upper bound, admits the minimal upper bound called 
(and, of course, the analogous statement holds for 
).![\varnothing\ne I_k=[a_k,b_k]\subseteq\mathbb R](https://s0.wp.com/latex.php?latex=%5Cvarnothing%5Cne+I_k%3D%5Ba_k%2Cb_k%5D%5Csubseteq%5Cmathbb+R&bg=ffffff&fg=000000&s=0 "\varnothing\ne I_k=[a_k,b_k]\subseteq\mathbb R")
is a family of nested nonempty closed intervals, 
, then the intersection 
is also nonempty.![f:[a,b]\to\mathbb R](https://s0.wp.com/latex.php?latex=f%3A%5Ba%2Cb%5D%5Cto%5Cmathbb+R&bg=ffffff&fg=000000&s=0 "f:[a,b]\to\mathbb R")
continuous on the closed segment ![[a,b]](https://s0.wp.com/latex.php?latex=%5Ba%2Cb%5D&bg=ffffff&fg=000000&s=0 "[a,b]")
, takes any intermediate value between 
and 
.
to 
. Everything you need to solve linear equations
and much, much more