What level of rigour is expected in Real Analysis? I fail to find a duplicate. 
I am wondering what level of rigour is needed in a typical undergraduate course in Real Analysis. To clarify my question, I provide an exercise from Rudin and my proposed solution:

(Exercise 5, Chapter 1) Let $A$ be a nonempty set of real numbers which is bounded below. Let $-A$ be the set of all numbers $-x$, where $x \in A$. Prove that $$\inf A = -\sup(-A)$$

My answer:

$A$ is bounded below. As such, $-A$ must be bounded above. Suppose $\alpha$ is the greatest lower bound of $A$.  It follows that $-\alpha$ is the least upper bound of $-A$. As such, we arrive at the desired expression $$\inf A = -\sup(-A)$$

This, for instance, feels very short, but I also feel that there is not much more to be said here. While this task might possibly be a bad example, I dare to guess that the most common pitfall for young students entering higher mathematics is that they underestimate the rigour needed to solve seemingly trivial problems. As such, I ask for an elaboration on this. The provided example does not necessarily have to be used in your answer.
 A: That is the expected answer. My experience with real analysis was questions that took a page or two followed by questions that follow as a simple corollary.
It is another common pitfall to assume that proofs have to be built up from first principles every time they are done.
A: Here's a proof along the lines that I would expect from an undergraduate student in first semester real analysis.
Proof: Let $A$ be a non-empty set of real numbers that is bounded below.  By definition of bounded below, we may choose $\alpha\in\mathbb R$ such that $\alpha\leq x$ for every $x\in A$.  This implies that $-x\leq -\alpha$ for every $x\in A$ so that $-\alpha$ is an upper bound for $-A$.  Thus, $-A$ is a non-empty set of real numbers that is bounded above and, therefore, has a supremum, say $\beta$, by the axiom of completeness.  
We must show that $-\beta$ is the infimum of $A$.  First, note $\beta$ is an upper bound for $-A$ (by definition of supremum) or $\beta \geq -x$ for every $x\in A$.  Thus, $-\beta\leq x$ for every $x\in A$ and $-\beta$ is a lower bound for $A$.  Next, we must show that $\beta$ is the greatest lower bound of $A$.  Thus, assume that $\beta<\gamma$.  Then, $-\gamma<-\beta$ so (since $\beta$ is the supremum of $-A$), there is some $x\in A$ with $-\gamma<x<-\beta$.  Therefore, $\beta<-x<\gamma$ with $x\in A$ so that $\gamma$ cannot be a lower bound of $-A$.$\Box$
To understand why these particular details are written out in grotesque detail, I would consider the material that you likely just learned.  If you are trying to show that an infimum can be defined in terms of a supremum, then you have likely just learned these concepts, as well as concepts like upper and lower bounds.  So I think you've really got to refer quite explicitly to those definitions.  By contrast, I used the order properties, like $x<y \implies -y<-x$ without specific reference since that's probably at least a little bit in the past.
A: $$
\inf A = -\sup(-A)
$$
Rudin's book gives definitions of the concepts involved, and I would stick close to what those definitions say.
$A$ is bounded below, i.e. it has a lower bound $x$.   That means $\forall a\in A,\ x\le a$.  Consequently $\forall a\in A,\ -x\ge -a$.
$\forall b \in -A\ \exists a\in A\  b = -a$, hence $\forall b\in -A,\ -x\ge b$.  Thus $-x$ is an upper bound of $-A$.
Thus we have proved that for every lower bound $x$ of $A$, $-x$ is an upper bound of $-A$.  In particular $-\inf A$ is an upper bound of $-A$.  In order to show that $-\inf A$ is the smallest upper bound of $-A$, one must show that no number less than $-\inf A$ is an upper bound of $-A$.  Suppose $c<-\inf A$.  Then $-c>\inf A$.  Since $-c$ is greater than the largest lower bound of $A$, $-c$ is not a lower bound of $A$.  Hence for some $a\in A$, $a<-c$, and so $-a>c$.  Since $-a\in-A$, we have a member of $-A$ that is greater than $c$, so $c$ is not an upper bound of $-A$. ${}\qquad\blacksquare$

I'd write something like that in an exercise in a section in which the concepts of upper and lower bounds and infs and sups were introduced.
