Inequalities:One key thing that many people seem to fail to grasp (or at least that I've noticed) is that you can use multiple inequalities to show one inequality, especially when your intuition tells you that the inequality is very loose.
So, for example - if I needed to show that
n!≥n (stupid example, but I can't think of any good ones right now) then you could say that
n!>2n (let's say it was in a question that was asking you to prove Sterling or some ****) then you could say
n!>2n>n. I'll see if I can think up some good example of this later.
Another thing to be aware of in STEP is to be very careful vis whether you're applying monotone functions to both sides of an inequality. So for example, it is true that (over the reals)
x3>y3⇒x>y but it is not true that (over the reals)
x2>y2⇒x>y since the
x↦x3 function is injective (but more importantly: increasing) over the reals and hence so is its inverse, but the
x↦x2 function doesn't have that same privilege.
If you're ever making a claim like
x3>y3⇒x>y in STEP, then make sure to say something like
x3 is increasing or such; or
x2>y2⇒x>y then say something like because
x,y>0 or
x2 is increasing over
R+.
A good way of establishing inequalities in STEP is to find a minimum or maximum and then show that the function is strictly decreasing or increasing from that point by considering the derivative over the interval. This comes in useful quite a bit as well.