What is the purpose of showing some numbers exist?
Solution 1:
Before I eat a sandwich it's a good idea to make sure it exists, right? Well, it obviously exists if I can see it, so I may not be aware of the need to establish existence, but I had established existence.
In other situations this becomes less silly, particularly in cases where the existence of whatever it is you wish to study or to use is not evident. Situations in real life are plentiful: Before you set out to study the supernatural you better establish that a supernatural phenomenon exists (famously, lots of research went into studying the supernatural without any established supernatural phenomenon in existence), before you believe the claims a book makes about the wishes of that or other god, you had better establish if god at all exists, before you use the psychic abilities of a medium in order to locate the whereabouts of a missing person, you had better establish the existence of the medium's claimed abilities. (James Randi dedicates a large portion of his life debunking such nonsense, all resulting from not being careful with checking that something exists before using it.)
In mathematics then it's not different. Before you can speak of the square root of $2$, you'd better establish that it exists. Similarly for other numbers. It's as simple as that. It's not a good idea to just accept that some number exists. After all, I can claim that there exists a real number $x$ which solves the equation $x^2+1=\rm {monkey}$, would you accept that? Probably not. What about my claim that there exists a real number $x$ solving $x^2+1=0$, do you accept that? Why not? So finally, do you just accept that there exists a real number $x$ solving $x^2-2=0$? Do you do that based on blind faith? Do you see the added value now in the proof that such a number actually exists?
A more purely mathematical answer: Some constructions of the real numbers lend themselves more easily to constructing certain numbers. The ability to construct a certain number in a given system may serve to show the utility of the construction. Thus, other than the general importance of establishing that certain numbers we expect to exist actually do exist, a particular proof of the existence of a number may be related to claims of the usefulness of a particular construction of the reals.
Solution 2:
You believe $\sqrt{2}$ is a number... so the question is whether or not it's a real number.
If the real numbers didn't have a number whose square is 2, that would be a rather serious defect; it would mean that the real numbers cannot be used as the setting for the kinds of mathematics where one wants to take a square root of $2$.
Basically, things cut both ways; while you could use the argument as justifying the idea of taking the square root of 2 is a useful notion, the more important aspect is that it justifies the idea of the real numbers is a useful notion.
Also, the argument is useful as a demonstrates of how to use the completeness properties to prove things.
Furthermore, it reinforces the notion that this type of reasoning can be used to define specific things. Some people have a lot of difficulty with this type of argument; e.g. "you haven't defined $\sqrt{2}$, you've just defined a way to get arbitrarily close to a square root of 2 without ever reaching it". It's a lot easier to dispel such misconceptions when the subject is something as clearly understood as $\sqrt{2}$.
Solution 3:
Consider the geometric questions:
Where does the line $y = x$ intersect the circle $x^{2} + y^{2} = 4$?
Where does the line $y = -1$ intersect the parabola $y = x^{2}$?
In 1., the existence of intersection points is equivalent to existence of square roots of $2$ in your number system. If your "universe" uses rational coordinates (which are visually indistinguishable from real coordinates), the answer is nowhere. Particularly, a line through the center of a circle in the rational plane $\mathbf{Q}^{2}$ need not intersect the circle. This is vexing for Euclidean geometry.
In 2., you may be inclined to answer "nowhere", but that would be a limited viewed imposed by your number system. Existence of intersection points here amounts to existence of square roots of $-1$.
The real point (heh) is, these questions are deeply analogous. The types of geometric construction you can make, and the types of solutions you can expect for systems of equations, depend very much on properties of your number system. If you're going to make sweeping claims (the intermediate value theorem, there exists a differentiable function equal to its own derivative, ...), you'd better have some theorems to back you up.
Solution 4:
There is one thing the existing (good) answers don't mention, which is what really the numbers that you are referring to are all about. It starts from the natural numbers, which are supposed to model counting up from 0 indefinitely. We really have no choice but to accept both their existence and properties on faith some way or another (see https://math.stackexchange.com/a/1334753/21820 if interested). But we can build a lot of things using just natural numbers. Firstly we can use a pair of natural numbers to denote fractions. Now what on earth are fractions? They are what happens when you want to consider division of things into smaller pieces or decomposing actions into identical subactions. This we denote using rational numbers. But in the real world, things don't seem to be quite rational at all. The Greeks had an ideal model of geometry that they used to describe the world, where points were infinitely sharp and lines were infinitely thin, and in that model at least there was a length that could be constructed by (ideal) straightedge and compass that could not be expressed as a ratio of two integers. As you probably know, that was $\sqrt{2}$. If the 'tools of perfection' (straightedge and compass representing circle and line) already produces irrational lengths, what more if we go beyond?
But there is an 'easy' answer. In the real world we only care that we can obtain a sufficiently good approximation of quantities. To measure a length of a table, we could use a ruler or measuring tape, and get an approximation to within 1m, or 1cm, or even 1mm. 'Clearly', to get better approximations all we need to do is to subdivide the measuring instrument markings into finer parts. Note that each approximation we make in this way is a rational fraction of a metre. That really is nothing more than the definition of a real number using decimal expansions, where getting a better approximation is simply taking more significant digits. All you have to do now is to define what addition and multiplication mean for decimal expansions and you would have reconstructed the field of real numbers. It then remains to prove all the basic properties that we had all been taught to take for granted, some of which are actually not so easy to prove. (It is now accepted that we cannot even measure any quantity to arbitrary precision, but that only matters at the quantum scale, and is way off-topic.)
So far we can get everything from the natural numbers (plus a few other things, like the existence of sequences of objects, equivalently functions on natural numbers, and so on). Of course, we should now check whether we really have got a real number whose product with itself is $2$. That is what your professor would have been going through the trouble of proving. Similarly for other numbers like $e$ that we use throughout mathematics. It cannot be very well appreciated what these results mean unless we really grasp the amazing fact that you can 'construct' a decimal expansion that when you multiply really has $2$ before the decimal point but all zeros after the decimal point! Did you realize how incredible that was? =)
Solution 5:
I don't think it's so much showing a number exists as it is showing the nature of real numbers is that all bounded sets have limits. If some numbers didn't exist we wouldn't really care whether $\sqrt 2$ did or not.
What we do care about is that if we have a set of rationals all of whose squares are less than $2$ and a set of rationals whose squares are greater than $2$, then it must be that some real has a square equal to $2$. That that must follow is what is of interest.