Big O Notation in two equations

Question

If $a = b + O (c)$, $d = e + O (f)$ and $b > e$, can we say that $a > d$? I proceeded by substracting the two equations. I think I have not done any thing wrong. It gives $a-d=b-e + O(c-f)$ and I am stuck here.Also c>f.

Answer 1

First recall that $$f=g+O(h)\iff \exists M>0,\quad |f-g|\leq M|h|$$ knowing this then it's clear that $b>e$ has not significance since we have $$a=b+O(c)\iff b=a+O(c)$$