In epsilon-delta proofs, when specifying delta, is it correct to specify the delta is equal to a value or delta is less than or equal to a value?
Which statement if more correct and why? Given epsilon > 0, choose delta = (number) OR Given epsilon > 0, choose delta < or = (number) OR Both?
Why? I would think that it would be more correct to specify that delta is < or = to a value since any smaller value would also work in the proof, right?
Thanks!
On
Let $A\subseteq \mathbb{R}, a\in A$ and $f:A\to \mathbb{R}$ be function. If you prove $$\forall \epsilon>0 \exists \delta>0 \forall x\in A (|x-a|<\delta \implies |f(x)-f(a)|<\epsilon) \ \ \ \ \ \ \ \ \ \ \ (1)$$ You can also prove that $$\forall \epsilon>0 \exists \delta>0 \forall x\in A (|x-a|\leq\delta \implies |f(x)-f(a)|<\epsilon) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$$ Because, If $(1)$ is true, there is a $\delta>0$ that satisfies $(1)$. If $\delta_{1}<\delta$, then $\delta_{1}$ satifies the $(1)$. Thus, we have found $\delta=\delta_{1}$ for $(2)$.
To show that $(2) \implies (1)$ is trivial one.
Hence, $(1) \iff (2)$.
You are correct when you say that "any smaller value would also work", at least in most common problems like limits, continuity, etc. But if we look at the definition for continuity of a function $f:\mathbb{R} \rightarrow \mathbb{R}$ at a point $a$, $$∀\epsilon>0\space \exists\delta>0 \text{ such that}$$ $$|x-a|<\delta \implies |f(x)-f(a)|<\epsilon,$$ note that we are saying that a $\delta$ exists. So it kinda doesn't matter (again, in most common problems) if it works for any $\delta$ less or equal than some value, but rather if there is some delta that works, which is normally the value you find for $\delta$ and show that works.