The $(ε, δ)$-definition of limit:For every real $ε > 0$, there exists a real $δ > 0$ such that for all real $x$, $0 < | x − a | < δ$ implies $| f(x) − L | < ε$.
From my point of view, I think ,in fact we want to make $f(x)$ can be made arbitrarily close to $L$ here, so it would be better to change “every real $ε > 0$” in the definition to "arbitrarily small real $ε > 0$", what do you think about my opinion here ?
That is exactly what we mean by "for every". We can take $\epsilon$ as small or large as we like. There is no need to change the statement, because it already hold for arbitrarily small $\epsilon$, since it holds for every $\epsilon \ge 0$. In math we want to be as general as possible, which here means saying that the statement must hold for every $\epsilon > 0$.