When proving limits, If I get $0<|f(x)-L|<10\epsilon$

Question

If I get $0<|f(x)-L|<10\epsilon$ canI say that the difinition of the limit holds?, can I say that the definition of the limit holds? My reasoning for this is that the goal of the proof is to prove that the statement is true for any arbitrary positive real number. Since I am dealing with real numbers, it shouldn't matter if I have 10 times epsilon, as that is also just an arbitrary positive number. I can get all the same numbers that I can with regular epsilon if I go small enough.

Answer 1

Yes.

You can argue formally like this. Given $\alpha>0$, you have shown there exists $\delta(\alpha)$ that makes $|f(x)-L|<10\alpha$ (for the relevant $x$).

Now, given $\epsilon>0$, choose $\delta(\epsilon/10)$.