I am trying to work through proving some sequences diverge. I am having a really hard time with the inequality arguments and I'm not sure why. The current problem is proving that $$\sqrt{n}-\frac{1}{n^2}+4$$ diverges to infinity.
I understand that essentially I let $c$ be an arbitrary positive number and then I have to find some natural number $N$ dependent on $c$ so that $$\sqrt{n}-\frac{1}{n^2}+4>c$$ for all $n\geq N$. The trouble I have is sussing out what $N$ needs to be for an arbitrary chosen $c$.
I think that I got it thanks to dem0nakos comment about not needing the best possible $N$.
Proof: Let $c$ be any positive number. By the Archimedean property we can select a natural number $N_1$ so that $N_1>4c^2$ and therefore $\sqrt{N_1}>2c$. Simultaneously we can find an $N_2$ such that $\frac{1}{N_2^2}<c$. If we let $N=\max\{N_1,N_2\}$ and take $n\geq N$ then we have $$\sqrt{n}-\frac{1}{n^2}+4>\sqrt{4c^2}-\frac{1}{n^2}+4=2c-c+4=c+4>c$$ as desired.