I'm looking for a verification of my $\epsilon-\delta$ proof of a limit example, if my proof is not completely mathematically rigorous, please tear it apart.

### Required to Prove:

$$\lim_{x \to 1} \ \ \frac{2+4x}{3} = 2$$

### Epsilon-Delta Definition

$$ \lim_{x \to a} f(x) = L \Leftrightarrow \forall \epsilon >0 (\exists \delta>0 \ni (\forall x \in D(0 < |x-a| < \delta \implies |f(x)-L| < \epsilon)))$$

### Proof:

I start out with $0 < |x-a| < \delta$ and use it to prove that it implies $|f(x)-L| <\epsilon$

$$0 < |x-a| < \delta$$ $$Let \ \ a = 1$$ $$0 < |x-1| < \delta$$ Multiplying by 4 throughout the inequality we get $$4(0) < 4|x-1| < 4\delta$$ $$0 < |4x-4| < 4\delta$$ Diving by 3 throughout the inequality we get $$\frac{0}{3} < \frac{1}{3}|4x-4| < \frac{4}{3}\delta$$ $$0 < |\frac{4x-4}{3}| < \frac{4}{3}\delta$$ $$0 < |\frac{4x+2}{3}-2| < \frac{4}{3}\delta$$ Letting $f(x) = \frac{4x+2}{3}$ and $L=2$, we can see that we have something similar to what we want to arrive at $$0 <|f(x)-L| < \frac{4}{3}\delta$$ This suggests that we choose $\frac{4}{3}\delta = \epsilon \implies \delta = \frac{3}{4}\epsilon$ $$\implies |f(x)-L|<\epsilon \ , \ \ \ \forall \ \delta =\frac{3}{4}\epsilon$$ $$\implies \delta(\epsilon) = \frac{3}{4}\epsilon $$

Therefore we have proven that given any $\epsilon > 0$, we can always find a $\delta>0$, that satisfies the inequality $0 <|x-a|<\delta \implies |f(x)-L|<\epsilon$

$$ Q.E.D.$$

**Some notes on my proof.**

*(and on writing delta as a function of epsilon)*

Since $\delta$ is dependent on $\epsilon$ I've written $\delta$ as a function of $\epsilon$ , i.e. $\delta(\epsilon)$. Sometimes this is seen as a "no-no" when writing $\epsilon - \delta$ proofs, because as a previous user put it :

"I guess one could also write something like $\delta(\epsilon)$ although implying that delta is a function of epsilon is in general incorrect because we can usually find multiple delta satisfying the condition for a given epsilon, not just one." - User A

"If for a given $\epsilon$ a $\delta$ is "good", then any $0<\delta'<\delta$ is also "good". So I'm not sure it'd be a well-defined function, but you can certainly think of it as one - keeping in mind that there are a lot of other $\delta$'s for that $\epsilon$." - User B

This implies that $\delta(\epsilon)$ is not a function as for any input ($\epsilon$) we have multiple output ($\delta$'s)

But since we are only trying to prove the existence of a single $\delta$ (to satisfy the $\epsilon-\delta$ definition of a limit), shouldn't writing $\delta(\epsilon)$ still be considered mathematically rigorous given the fact that we are only finding a function $\delta(\epsilon)$ that has a co-domain of a subset of all possible $\delta$'s? If not what are possible ways to workaround this, and introduce further mathematical rigor to this last step?

So is my proof mathematically rigorous? Are there any errors I've made? If this proof is not completely mathematically rigorous, please tear it apart, I'm looking to improve my proof-writing skills as an undergraduate student majoring in Pure Mathematics, so please feel-free to offer any criticism and/or suggestions if you feel necessary.