Proving Big Omega of a polynomial without limits

Here is the definition of $ \Omega$ :

$ f(n) = Ω(g(n))$ iff there exist positive constants $ c$ and $ n_0$ such that $ f(n) \ge cg(n)$ for all $ n\ge n_0$ .

Here is one theorem:

If $ f(n) = a_m n^m + \cdots + a_1 n + a_0$ and $ a_m > 0$ , then $ f(n) = \Omega(n^m)$ .

I want to prove this, without using limits. Despite many hours of searching across the internet, all I could find is proofs using limits. Is there any other way?