Question

How to prove this:

x^7 = O(x^10)
x^10 = O(x^7)?

I couldn’t prove this statement.

Answer 1

Let’s take a look at the definition of big-O notation.

f ∈ O(g) <=> (∃ x) (∃ c > 0) (∀ y > x) (|f(y)| <= c⋅|g(y)|)

The right hand side can be formulated “the quotient f/g is bounded for sufficiently large x“.

So to prove that f ∈ O(g), look at the quotient, choose a (largish) x and try to find a bound.
For the first case, the quotient is

x⁷ / x¹⁰ = 1/x³

A bound for x ≥ 1 is obvious.

To refute f ∈ O(g), look at the quotient and prove that it assumes values of arbitrarily large modulus on each interval [x, ∞). Assume an arbitrary c > 0, and prove that for any x, there is an y > x with |f(y)/g(y)| > c.

That should give enough of a hint.

If not: x³ > c for x ≥ c+1.