How to express a sentence having two for all?

Question

I want to write a sentence in a journal but the sentence has two 'for all's.

For example,

The variable $x$ is the smallest number such that $f(\zeta x) \le g(a)$ for all $\zeta\in[0,10]$ and for every $a\in A$.

Is it fine if I write like the above?

Another cases are

The variable $x$ is the smallest number such that $f(\zeta x) \le g(a)$ for all $\zeta\in[0,10]$ and $a\in A$.

The variable $x$ is the smallest number such that, for all $\zeta\in[0,10]$, $f(\zeta x) \le g(a)$ for every $a\in A$.

What is the best expression among them? If you know better expression, please let me know. Thank you.

Answer 1

The key to good writing is clarity and simplicity. One way to achieve this is to group things together. Consider the following format:

"... such that [list of statements] for every [list of definitions]."

As we naturally like to group things, using this format which groups definitions and statements together makes it easy for us to understand what it is conveying.

To answer the above questions, All three expressions are clear and okay to use. Of the three expressions above, the second one is the easiest to understand.

Answer 2

Your second sentence is not so good as it is more asymmetric and will make people thing that the status of $\zeta$ and $a$ differ. The first is fine.

If you want to stress dependency on two variables, you can also write

The variable $x$ is the smallest number such that $f(\zeta x) \le g(a)$ for all $\zeta$ and $a$, subject to $\zeta\in[0,10]$ and $a\in A$.

Answer 3

Are you sure you really mean this? If we are talking only about nonnegative $x$ here then the smallest $x$ satisfying your condition (if there is such an $x$ at all) is $0$.

You could save one "for all" by writing "Let $x_*$ be the smallest (?) $x\geq0$ satisfying $$f(\zeta x)\leq \inf_{a\in A} g(a)$$ for all $\zeta\in[0,10]$."