TITLE: When learning mathematics should one prove everything one learns?
QUESTION [14 upvotes]: I sat through a real analysis class around a year ago and in about two days we partially covered the construction of the real numbers as equivalence classes of Cauchy sequences. Through the teacher didn't do it, it took me about $9$ hours to read and then write out the entire construction in a way I understood it, starting from $\mathbb{N}$. Most of the process was laborious verification of algebraic manipulations and just checking certain things were satisfiable. Despite doing all of that, I don't think I gained any particularly new insights, it was just a lot of work. 
At what point should you not verify something? What if you can see there is a proof of it, and you see all the proof requires is verifying a huge amount of algebraic manipulations? In this case even if you go through and check the proof you can be positive manipulating algebraic expressions won't teach you anything new. So then why bother? Where does one draw the line saying  "I should read a proof of this" vs "there isn't anything to be gained here"?

REPLY [2 votes]: I'm afraid you can't get away without proving things in mathematics (although you certainly don't have to verify every detail of every proof that you read). You have a good point that the usual constructions of the real numbers are often presented as if the verification of the required properties is an easy exercise, whereas the very readable and detailed account in Landau's Foundations of Analysis (using Dedekind sections) occupies over 90 pages. To quote John H. Conway (On Numbers and Games p. 26):

In practice the main problem is to avoid tedious case discussions. [Nobody. can seriously pretend that he has ever discussed even eight cases in such a theorem [the construction of $\Bbb{R}$ from $\Bbb{Q}$ via Dedekind sections or Cauchy sequences] - yet I have seen a presentation in which one theorem has 64 cases.]

On the other hand, many people (myself included) have carried out complete formal verifications of such constructions with some computer assistance, in systems such as HOL. But even with machine support, it is good to use nicer methods than crude case analysis. One simple observation is that once you have shown that the additive structure $(\Bbb{R}, 0, +, <)$ is a complete totally ordered abelian group, you can define and verify the multiplicative structure by studying the order-preserving homomorphisms of $(\Bbb{R}, 0, +, <)$, which is both more interesting and less error-prone than working directly with Cauchy sequences or Dedekind sections.
Schanuel's Eudoxus Reals provide another entertaining alternative construction of the real numbers involving an interesting investigation of self-mappings of the integers that aren't quite additive homomorphisms.