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 $$\inf A = -\sup(-A)$$
$A$ 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.