If we want to show something for every $x \in X$ of, sometimes we simply say: Let $x$ be $\in$ of $X$ (x is arbitrary). And if we continue doing that,after the proof is done, this allows us to say that it holds $\forall x$. Why does it work ? I mean it makes somehow sense, as $x$ is arbitrary and we havent said anything for $x$ apart from that $x$ is an element of $X$. Thanks
EDIT: Someone asked for an example. A simple example would be if $n\in \Bbb N$ and $n$ is even, then $n^2$ is even. So we could start by saying let $n$ be an arbitrary natural number, which is even. There exists a $k\in \mathbb(N)$ such we can say $n=2k$. $(2k)^2=2*2k^2$ which is even. So we have simply taken an arbitrary natural number, which is even, and concluded that it is valid for all numbers with the same property
Suppose I have a set $X = \{1,2,3\}$ and I want to show each element in $X$ satisfies some property $P$. One method of attack for this proof is to start with the element $1$ of $X$, and show that it satisfies $P$. Then I could move on to element $2$ of $X$, and show that it satisfies $P$. Finally, I can show $3$ satisfies $P$.
As I am writing the above proofs, I notice that the steps in the proofs for each element are all the same. So I really just did three times the work when writing my proof. I could have instead said "Let $x \in X$ be arbitrary" and then did the steps on the arbitrary $x$. This would take care of all of the elements of $X$ at once because, by letting $x$ be arbitrary and showing $P$ holds for $x$, I am saying the proof works for every element of $X$. Does that make sense?