Which is it, ceiling or floor? Part 1

For the floor and ceiling functions, for $n\in\mathbb{N}, -1<\epsilon<1$ , we have that $\lfloor n+\epsilon \rfloor=n+\lfloor\epsilon\rfloor$ and $\lceil n+\epsilon \rceil=n+\lceil\epsilon\rceil$ .

Let's work our way from the inside out:

First of all, $\lceil2.5\rceil=3$

Next, $\lfloor3+0.1\rfloor=3$

Next, $\lceil3-0.2\rceil=3$

See a pattern? We can use the property above to prove it:

• First, we rewrite $\lceil\lfloor...\lceil\lfloor3+0.1\rfloor-0.2\rceil+0.3\rfloor-...-1\rceil$ as $\lceil3+\lfloor0.1\rfloor+\lceil-0.2\rceil+...+\lfloor0.9\rfloor-1\rceil$

• The floor sends a number between 0 and 1 to 0 and the ceiling sends a number between -1 and 0 to 0, so the expression simplifies to $\lceil3+0+0+...+0-1\rceil=\lceil3-1\rceil=\lceil2\rceil=2$