5 \frac{1}{n} - 2 \frac{1}{n^2} | < \frac{5}{n}+\frac{2}{n^2}You'd have to ask him/her. It's not really ideal, but it's not indefensible (if you use non-strict inequality signs).
And yes the inequality should be strict, but for values greater than 0, then LHS = RHS for all values as OP had written
No, it's precisely when you may have that LHS = RHS that you cannot say that LHS is strictly smaller than the RHS.
So if they had used a + sign s.t. the end of their first line read:
<= (5+n) / (2/n^2) then would that be more ideal than how OP posed it?
Assuming you mean (5+n)/(2n^2), not hugely. You generally want to do two things when saying "A < B" in part of an analysis argument. First you should make sure than it's true that "A < B", but secondly, if you're then going to be working with B, you want to try to make sure B is something nice to work with. (5+n)/(2n^2) passes the first test but not the second.
Probably the cleanest approach here is to go:
for n >= 1, 0 < 2/n^2 < 5/n, so 0 < 5/n - 2/n^2 < 5/n, and then given epsilon > 0, we have n > 5/epsilon => 5/n < epsilon.
But to be honest it's hard to definitively mess this question up - even if you omit justification (as I'd guess is what happened here in the original post), the justification is so trivial that you're likely to get the benefit of the doubt unless someone's being incredibly picky.