Big-Oh of polynomials

Definition of O(n²)
Basically, f(n) is in O(n²) iff ∃c∈R⁺, ∃B∈N, such that ∀n∈N, n≥B ⇒ f(n)≤cn²

In english, the statement is saying that there exists a number i that is greater than B (breakpoint) such that f(n) is upper bounded by cn².

Now when proving a polynomial is upper bounded by O(n²), it is all about picking c and B.
- The first thing is to pick a c which should be larger than than the constant factor of the highest-order term in the polynomial. - Next is to pick a B, usually B = 1 works in most cases but should be careful when doing so.

For an example, 7n⁶-5n⁴+2n³ ∈ O(6n⁸-4n⁵+n²)
Some things to consider or make assumption is to:
1) assume n ≥ 1
2)Upper bound the left side by overestimating
2)Lower bound the right side by underestimating
3)Finally, choose a c that connects the two bounds.

7n⁶-5n⁴+2n³ ∈ O(6n⁸-4n⁵+n²)
pick c = 9/2
pick B = 1
     assume n∈N
         assume n≥1
             then 7n⁶-5n⁴+2n³ ≤ 7n⁶+2n³      #-5n⁴ ≤ 0
                                           ≤  7n⁶+2n⁶ = 9n⁶      #n³ ≤ n⁶since n ≥ 1
                                           = (9/2)2n⁶ = c2n⁶     # c = 9/2
                                           ≤ c2n⁸ = c(6n⁸ - 4n⁸) ≤ c(6n⁸ - 4n⁵)     # -n⁸ ≤ -n⁵
                                           ≤ c(6n⁸-4n⁵+n²)     # 0 ≤ n²
            then 7n⁶-5n⁴+2n³ ≤ c(6n⁸-4n⁵+n²)
 <introduce the antecedent and quantifiers and you are done!>

I think the biggest advice before tackling such proofs is to first set  6n⁸-4n⁵+n² as the largest(which the polynomial be upper bounded to) and set the 7n⁶-5n⁴+2n³ (the polynomial to be bounded by O(n²)) to be the lowest. Afterwards you slowly decrease the upper bound and increase the lowest and meet halfway  by choosing a c that connects the two bounds.