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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?

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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.)

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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".

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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)

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