Here all the functions of natural numbers are positive.$p$ is a prime number. Let $A(n)+B(n)=C(n)+D(n)$ where $B(n)=O(p^{5n})$. $ A(n)$ is always less that or equal to $C(n)$.$D(n)$ is known to be of $O(p^{4n})$. Can I deduce that $C(n)$ is greater or equal to $B(n)+O(p^{4n})$ ?
I arrived in this kind of equation by the following way: I have an exact sequence of vector spaces $V(n),W(n),P(n)$ over a field $K$.
$V(n)->W(n)->P(n)->0 $ .I complete this exact sequence into $0->M(n)->V(n)->W(n)->P(n)->0 $. Here M(n) is the kernel of the map from $V(n)$ to $W(n)$. Also $A(n), B(n), C(n), D(n) $ are dimension of the vector spaces $M(n),W(n),V(n),P(n)$ respectively. I apply the dimension formula for exact sequence and get the given equality.