Why should we prove obvious things?
Because sometimes, things that should be "obvious" turn out to be completely false. Here are some examples:
- Switching doors in the Monty Hall problem "obviously" should not affect the outcome.
- Since Gabriel's horn has finite volume, then it "obviously" has finite surface area.
- "Obviously" we cannot decompose sphere into a finite number of disjoint subsets and reconstruct them into two copies of the original sphere.
- Since the Weierstrass function is everywhere continuous, then "obviously" it must have at least a few differentiable points.
Of course, mathematics has shown that switching doors is to the player's advantage, that Gabriel's horn actually has infinite surface area, that you can indeed get two copies of the original sphere (see Banach-Tarski paradox), and that the Weierstrass function is everywhere continuous but nowhere differentiable. The point being, there are many things out there which are "obvious" but actually turn out to be entirely counterintuitive and opposite what we would otherwise expect. This is the point of rigor: to double check and make sure our intuition is indeed correct, because it isn't always.
I think the answer has four parts.
If you ask a random person at Walmart what $$\lim_{n\to \infty} \frac{1}{n}$$ is, then you might not get much. If you tell them that it is $0$, then they probably won't think that it is obvious.
Conversely, if you go to a high level research talk, you will hear "It is obvious that ..." or "It is clear that ..." a lot. And you might not think that it is very obvious.
The point is: Whether something is obvious or not is relative to the person.
Mathematics is built around proving things. There is a justification for everything. This is the very nature of mathematics. We start with some axioms and then we prove everything. So when you ask, "Why should we prove something?", the answer always contains: "Because we are doing mathematics."
You don't need much experience teaching mathematics before you meet a student who is confused about losing points on an exam because of lack of justification. Often the student will respond that they just thought that it was obvious. When you press them a bit harder it becomes clear that they, in fact, have no idea how to justify what they did. Whether or not the student arrived at the correct answer is irrelevant; the point is: if something is truly obvious, then it shouldn't be hard to prove it.
If you want to get good at proving difficult things, why not get experience with proving things by starting to focus on simple or "obvious" things? I think that the experience gained from proving even simple propositions is valuable later in your career as a mathematician.
The main procedural reason is to show that your axioms correctly capture what you want them to capture: that is to say they are both "correct" and sufficient.
If it turned out that under our axioms $\lim\limits_{n\to\infty}\dfrac{1}{n}\neq0$ then we would probably choose a different definition of $\lim$ (or a different name for it), since it would not be describing anything that we'd like to call a "limit". It would not be "correct". Of course that wouldn't be a problem in some unusual topology, since by calling it "unusual" we mean that we don't expect it to behave the same as the usual one, so limits might be different. There may be a fine line between a result that's counter-intuitive but that we stand by our system anyway, and a result that causes us to conclude that our definitions or axioms aren't as useful as we thought they were.
Consider that Euclid tried and failed to prove the "obvious" parallel postulate. Fast forward 2000 years or so, and it's finally proved not to be a theorem of Euclid's other axioms. His first four axioms were not sufficient to describe what was "obvious". Furthermore, non-Euclidean geometries (in which the postulate is not true) are interesting and useful.
It is valuable to know whether or not "obvious" things are provable from your axioms.
When learning mathematics, it's useful to prove "obvious" results in addition to "non-obvious" ones because:
- you "know" they're true before you start, which can save some frustration
- the ease or difficulty of proving the obvious teaches you something interesting about the area you're working in
- you train yourself to reason only using formal axioms, not by assuming any old "obvious" things you like, no matter how tempting they are
- similarly you train yourself to accept from others only things that are proven, no matter how true they look
- probably other benefits.
Then when something is stated as "obvious", or you want to state it so yourself, you quickly either prove it to yourself, or at least satisfy yourself that a proof is possible and you could write it out if really needed, or else you question the "obvious". It might turn out to be false (in which case you've avoided an error) or it might turn out to require quite a difficult proof (not so obvious after all despite your intuition being correct). Normally you would want to restrict the use of the word "obvious" to things where the first proof your reader would think of works (and hence anyone can easily prove them if they bother to write it out), not to things where your intuition is correct but the proof is tricky.