Show that a function is a Borel function

Question

Let $u:\mathbb R \rightarrow \mathbb R, ~ u(x) = \text{ceil}(x)$ (i.e. round $x$ up to nearest integer).

I need to determine

1. What is the set $\{ u \leq a\}, ~~~ \forall a \in \mathbb R$

2. Show that $u$ is a Borel function i.e. that it is $\mathcal B(\mathbb R)/\mathcal B(\mathbb R)$

My thoughts so far:

1. The sets will be all sets that can be written on the form $\{-\infty, \dots ~, \text{floor}(a)-2, ~\text{floor}(a) -1,~ \text{floor}(a) \}$. As we let $a$ run to $\infty$ we end up with the set of all integers.
2. if I show $\{ u \leq a\}\in \mathscr A$, (where I think $\mathscr A$ should be $\mathcal B(\mathbb R)$?) then $u$ should be $\mathcal B(\mathbb R)/\mathcal B(\mathbb R)$. Is this true, and if so how do I show this mathematically?

Or is it better to show using the pre-image, and if so how? e.g. similar to this (problem is I don't quite understand it)

Answer 1

$\{u\le a\}$ indeed denotes the preimage $u^{-1}((-\infty,a])=\{x:u(x) \le a\}$. This is going to be $(-\infty, \lfloor a\rfloor]$ which is an element of $\mathcal B(\Bbb R)$.

For your second example, $v:\Bbb R^2\to\Bbb R$, so it's rather the single $(-\infty, a]$ rays that generate $\mathcal B(\Bbb R)$, and the '?' is to be filled by $a$.
The preimage of $(-\infty, a]$ would be a disk with $v'(x)=\lceil x^2 +y^{\bf 2}\rceil$, but it's a closed set also for $v$, with boundary $x^2+y^3=\lfloor a\rfloor$.