When to use For any vs Let at the start of a proof? I was wondering when to use "for any"; vs "let...be..." at the start of a proof?
For example,  a proof may start like this:
For any $a$ invertible in F, there exists  $a^{-1}$ such that  $ a^{-1}\cdot a =1. $ Then .....
vs.
Let  $a$ in F be an invertible element, there exists $ a^{-1}$ such that  $ a^{-1}\cdot a =1. $ Then .....
How do I decide which one to use? and what are the differences?
 A: You can generally use either.
"For any" means that the following proof will hold for any $a$, and the grammar is most naturally completed with a claim.
"Let $a$ be..." just specifies the properties of variable $a$.  One need not make a claim at the end of the sentence.  It might be used in a claim later.  "Let $a$ be an integer, $b$ be a real positive number, and let $c$ be a complex number."  (No claim is made.)
A: It mainly depends on the next sentence. But, first let me observe that the punctuation of the second version is incorrect and the comma should be a dot:

Let $a$ be an invertible element in $F$. By definition, there exists $a^{-1}$ in $F$ such that $a^{-1}a = 1$. Then ...

Now, back to the question. You could use the "For any" version if you don't use $a$ in the next sentence, like in "Therefore, $0$ is not invertible". On the contrary, if you reuse $a$, you should use the second version, like in "It follows that, for every integer $n$, $a^n$ is also invertible".
A: let: If you are going to describe a general element with a specific property. (let $a\in\mathbb{N}$ s.t $a$ is multiple of $2$).
for: If this is element is very general. ( For any vector space $V$, we can find a basis)
