Let: and for some .
Fact: where
Consider:
Take the floor function of both sides and with the fact, the conclusion follows.
I think this is supposed to be . A quick counterexample to what you have written is .
So let . So , and since we can conclude by the defintion of the floor function that . Now lets look at the left side. Defining the same as above we can see that and once again because we can conclude that . Finally
Similar to before define . So once again by the definition of the floor function . SoShow that
Now for the second part notice that . Now conclude the answer by noticing that since