Let $A$ be a set. I want to prove that the set is infinite. I have used the proof by contradiction. First of all I have noticed that the set is not empty and after that I have assumed that the set is finite and then I get a contradiction.
My question is: Is the assumption the set is not empty really needed for my proof or one can omit this step.
On
An empty set is usually considered to be finite.
However if your "it can't be finite" proof actually only proves "it can't be nonempty finite" (for example because you need to pick an element from it as part of the proof or you're using an induction that can only start at $1$), then you'd still need a separate "it can't be empty" step.
Call $X = \{n \in \mathbb{N}| n < 0\}$ (yes, the negative natural numbers!). You can prove from the axioms of the natural numbers that there is no natural number $n$ such that $n+1=0$. So if $n<0$, then $n+1<1$ and $n+1\neq 0$, so $n+1 < 0$. So if $\exists n \in X$ then $X$ is infinite!