Proving an inequality by rearranging it

Question

I want to prove the statement: $∃a∈ℕ,∀x∈ℕ,x>a⇒\frac{x^5}{50}≥-x^4+5$

I learned that you can't assume the conclusion in the beginning when proving but can I still rearrange the inequality to prove something else? For example:

$\frac{x^5}{50}≥-x^4+5$ can be rearranged to $x^5+50x^4≥250$

So instead, I will prove $∃a∈ℕ,∀x∈ℕ,x>a⇒x^5+50x^4≥250$

Let $a=10$

Let $x∈ℕ$

Assume $x>a$

$x^5+50x^4=x^4(x+50)>10^4(10+50)=600000≥250$

Is something like this allowed?

Answer 1

yes, that is allowed.

You have used the property that $x^5+5-x^4$ is an increasing function and also

$$\frac{x^5}{50} \ge -x^4+5$$ is equivalent to $$x^5+50x^4 \ge 250$$

Alternatively, notice that over the positive domain, $f(x)=\frac{x^5}{50}$ is an increasing function and $g(x)=-x^4+5$ is a decreasing function.

In particular $f(2) > 0 > g(2)$, hence for all $x > 2$, $f(x) \ge g(x)$.