# 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?

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.)

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".

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)