$f_1$, $f_2$, $g_1$, and $g_2$ are functions such that:$$f_1 = \Theta(f_2)$$$$g_1 = \Theta(g_2)$$I was able to prove that:$$\frac{f_1}{g_1} = \Theta\biggl(\frac{f_2}{g_2}\biggr)$$But I can't understand why the following statment is not necessarily true:$$f_1 - g_1 = \Theta(f_2 - g_2)$$
Why do -bounds not survive taking differences?
asymptotics;landau notation