When stating a theorem in textbook, use the word "For all" or "Let"? [duplicate]

If you're doing informal mathematics, there's really no difference. I guess from a type-theoretic perspective, it's kind of the difference between $$x:\mathbb{R} \vdash P(x) \qquad \mbox{and} \qquad \vdash (\forall x:\mathbb{R})\,P(x).$$

The former is arguably better, because it doesn't presuppose that we're working in a category that interprets universal quantification. So "let" is preferable to "for all" for this reason. But, again, unless you're doing highly formal mathematics, it's not really worth worrying about. (I say that, but a part of me finds the question fascinating, and I've just gone and favourited it.)

In your first example, either is fine. In your second example, 2 is ungrammatical - you cannot just replace "Let" with "For all", the "be" has to be deleted (or replaced with "that are"). Otherwise, either is fine.

In all three examples, you'll notice that your "For all" example results in a long, slightly awkwardly-phrased sentence with at least three clauses. The "Let" version divides the sentence into two simpler sentences, so the reader can process it one step at a time.

In general, "For all" is okay as long as the thing you're quantifying over is small and doesn't really require any work to understand. If the reader's going to have to think about it even a little - e.g. "system of linear equations in five variables" - I'd use "Let".

Also - and I'm pretty sure this is just a personal preference - I try to avoid having more than two parts to a sentence in a mathematical theorem, if I can. "If $X$ then $Y$" is fine, but "If $X$ then if $Y$ then $Z$" is ugly. Similarly, "For all $x$, if $Y$ then $Z$" is complicated, and it gets worse the more complicated $x$, $Y$, and $Z$ are.

As a personal point of view I would first "fix" what I'm working with with a let, and then state my property with for all if needed.

So my canonical form would be:

Let a be something, b be something, and c be something. If for all d such that something on a,b,c and d then we have something great on a,b,c

I think it is the more readable way to state a theorem.

So to go through your examples:

Example set 1.

Let $n$ be a natural number. If $n$ is even, then $n$ squared is even.

Example set 2.

Let $A,B$ be two sets. If for all $x\in A$, we have $x\in B$, then we say that $A$ is a subset of $B$.

Example set 3.

Let $Y$ be a subspace of $X$. $Y$ is compact if and only if every covering of $Y$ by sets open in $X$ contains a finite subcollection covering $Y$. (Munkres Topology Lemma 26.1)

Example set 4.

Let $f:X\to Y$ be a bijective continuous function. If $X$ is compact and $Y$ is Hausdorff, then $f$ is a homeomorphism. (Munkres Topology Theorem 26.6)

** example set 5**

Let $E,f:E\to\mathbb{R},L,c$ be things. If for all $\varepsilon>0$, there exists $\delta>0$, such that for all $x\in E$, $0<|x-c|<\delta\rightarrow |f(x)-L|<\varepsilon$", then we say $f(x)$ converges to $L$ when $x$ approaches $c$.

Some of this is a matter of style - which isn't very important. But there are a few "technical" problems with some of your examples.

Examples 2.1 and 2.2:

If for all $x \in A$, $x \in B$, then ...

If you parse this as "If ( for all $x \in A$ and $x \in B$ ) ..." it doesn't make sense. The intended meaning is "For all $x$, if $x \in A$ implies that $x \in B$, we say ...".

Since you are considering the $x$'s "one at a time" here, I would prefer "for each/any/every $x$" to "for all $x$".

Example 3.2:

I would prefer "for each/any/every subspace" to "for all subspaces", for the same reason as above.

Examples 4.1, 4.2

The grammar seems a bit convoluted here. The main statement is of the form "If (the premises) then (the conclusion)" but you have a subsidiary clause (stating the properties of $X$ and $Y$) after the premises. I would re-order 4.1 as something like

If $X$ is compact and $Y$ is Hausdorff, then every bijective continuous function $f:X \to Y$ is a homeomorphism.

and similarly for 4.2.

Example 5.3

"If let $\epsilon > 0$ ..." is not correct English grammar. You need a clause after the "If", but "let $\epsilon > 0$ ... then $|f(x)−L|< \epsilon$" is a complete sentence, not a clause within the bigger sentence starting with "If".