Inequality Problem $\forall A , B,N>0\exists C>0:Cn^4\geq An^3 + Bn^2 + N$

Question

Okj guys I need some Tipps with this one. Tried it with Induction, but just cannot figure it out

$\forall A , B,N>0\exists C>0:Cn^4\geq An^3 + Bn^2 + N$ (C has to be independent from n)

Answer 1

Given $A,B,N>0$,

we have

$\lim_{n\to \infty}\frac{n^4}{An^3+Bn^2+N}=+\infty$ thus for enough great $n$, we will have

$\frac{n^4}{An^3+Bn^2+N}>1$ for example.

so, we can take $C=1$. there are many possibilities.

Answer 2

for $A,B,N>0$ and $n \geq 1$, we have

$An^3\leq An^4$

$Bn^2\leq Bn^4$

$N \leq Nn^4$

so $An^3+Bn^2+N \leq (A+B+N)n^4$

thus, we can take

$C=A+B+N$.