How can it be formally proved that $f \in O(⌊f ⌋)$

Question

I'm trying to prove that $f \in \mathcal{O}(\lfloor f \rfloor)$ given that $\forall m \in \mathbb{N}, f(m) \geq 1$

Here's what I've thought of so far, we can set C = 10 and k = 1 and somehow prove that this would eventually dominate the function? Not sure how I can prove this without a concrete function defined for $f$

According to the definition of big O,

For any C larger than 1, C times floor of f would eventually dominate f. What I can't figure out is how to outline this in a formal proof. Any help is appreciated!

Answer 1

$$\{f\}<1\le\lfloor f\rfloor\implies f=\lfloor f\rfloor+\{f\}<2\lfloor f\rfloor.$$

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Together with $2\lfloor f\rfloor\le2f$, that makes $f=\Theta(\lfloor f\rfloor)$.