Do “for any fixed $x$” and “for arbitrary $x$” differ in meaning? 
Let $f(x,y)$ be a continuous function in $x>1/2$ and $y>3.$
Statement 1: For any fixed $x>1/2$ $$|f(x,y)|\leq C y  \ \ \ \ \text{whenever} \ \ \ \ y\geq y_0$$ where $y_0$ is fixed and $C>0$ is a constant.
Statement 2: For arbitrary $x>1/2$ $$|f(x,y)|\leq C y  \ \ \ \ \text{whenever} \ \ \ \ y\geq y_0$$ where $y_0$ is fixed and $C>0$ is a constant.

My question: Does statement $1$ imply statement $2$? Why do we sometimes use the first statement instead of the second statement?
 A: 
Statement 1: For any fixed $x>1/2$ $$|f(x,y)|\leq C y  \ \ \ \ \text{whenever} \ \ \ \ y\geq y_0$$ where $y_0$ is fixed and $C>0$ is a constant.
Statement 2: For arbitrary $x>1/2$ $$|f(x,y)|\leq C y  \ \ \ \ \text{whenever} \ \ \ \ y\geq y_0$$ where $y_0$ is fixed and $C>0$ is a constant.


*

*When constructing a proof, we might begin by writing

*

*Take/Consider any (i.e., an arbitrary) value of $x$ greater than $\frac12$.

The idea is that this value is fixed when applying the remaining steps, and every time we reiterate the proof's flow of logic, we are free to arbitrarily choose any value for $x$ and fix it for that iteration.
After we have reached its conclusion, we might summarise the proof, in other words, write its theorem statement, by using Universal Introduction to convert the above opening sentence to

*

*“For each $x{>}\frac12\ldots$”

So, in each of your given statements, the boldfaced phrase is more clearly and accurately replaced with “each”. And since “for $x{>}\frac12$” implicitly means “for each $x{>}\frac12$”, the boldfaced phrases are in fact superfluous.


*The actual problem with your given statements is the potential ambiguity of the word ‘where’: does $$P(x)\text{ is true where }Q(x)\text{ is true}$$ mean $$Q(x)\text{ implies }P(x)$$ or $$Q(x)\text{ and }P(x)\;??$$


*Here's the best (and safest, given the above ambiguity) rephrasing:

*

*For each $y_0,$ each $C{>}0$ and each $x{>}\frac12,$ $$\text{if } \ \ y\geq y_0,  \ \ \ \ \text{then} \ \ \ |f(x,y)|\leq C y.$$
