Understanding Asymptotic Notation of a constant

Question

How can I prove that if $f(n) = O(1)$ leads to $f(n) = \Omega(1)$ as well?

I need a Formal definition of the meaning that a function $f(n) = O(1)$

Answer 1

The formal definition of $f(n)=O(1)$ is that there exists $M$ such that for all sufficiently large $n$ (for it is implicit that we consider $n\to\infty$) $|f(n)|\le M|1|$. If $f$ is only defined on $\mathbb N$ anyway, this is equivalent with: $f$ is bounded.

The Hardy-Littlewood definition of $f(x)=\Omega(1)$ would mean that $\limsup_{n\to\infty}\left|\frac{f(n)}{1}\right|>0$. Note that with this definition, $f(n)=O(1)$ does not imply $f(n)=\Omega(1)$: Just consider $f(n)=\frac1n$. Knuth uses a stronger definition, namely that $1=O(f(n)$ - which does not follow either.