Is it true that for all real numbers, there exists an integer smaller than it?

Question

I cannot conclusively answer this as I have two different opinions:

1. It is false considering that the set of integers is a subset of the reals and that the “smallest” integer is also a real, hence there is a real number which does not have a smaller integer
2. It is true, considering that there are infinite integers, so there is always an integer smaller than a real

Could someone explain this to me?

Answer 1

Let $x \in \mathbb{R}$.

If $x \in \mathbb{R}$ but $x \notin\mathbb{Z}, [x]$ is an integer smaller than $x.$

If $x \in \mathbb{Z}, x-1$ is an integer smaller than $x.$