A common mistake was to start with the inequality to be proven, and then finish with a true statement such as x^{2} ≥ 0. The way to prove something is always to start from a true statement and finish with what is to be proven.
The only reason the other approach "seems to work", is because the steps taken can all be reversed, but this has to be made explicit (and justified) .
Your textbook has more explanations on that particular mistake, p. 43. (The section right before the problems usually contains very valuable advice...).
A few students did not think about the ambiguity raised by the number of days in the month of February.
To prove equality between two sets, it is not enough to prove one is a subset of the other. This is also (especially) true if one is just giving an explanation for the equality in plain English.