For all real numbers $x$, prove $\lfloor x - 2\rfloor = \lfloor x\rfloor - 2$

Question

Prove the following statement:

For all real numbers $x$, $\lfloor x - 2\rfloor = \lfloor x\rfloor - 2$

I'd appreciate some help with this.

All I know is that the floor function $n$ implies : $n \leq x < n+1$

Answer 1

Let $x$ be a real number. Then by the well-ordering principle there exists a unique $n\in\mathbb{Z}$ such that $n\leq x<n+1$.

This implies that $\lfloor x\rfloor = n$ by definition of the floor function.

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From here, you need to come up with an argument for why $\lfloor x-2\rfloor = n-2$.

Does it follow from the above that $n-2\leq x-2$?

Does it follow from the above that there is no integer larger than $n-2$, lets call it $m$, such that $n-2<m\leq x-2$?