"Closeness" Definition writing for sets I am having trouble writing the definition for this example:
The closeness of the set {2,3,5} to 0 is 2. The closeness of the set {1.31,1.301,1.3001..} is 1.3. The closeness of the set (3,5] to 0 is 3. 
So, I concluded that I cannot write a definition about closeness based on what number is in the set because the closeness of the set (3,5] is 3, but 3 is NOT in the set. I am not sure how to approach this definition. Can anyone give me some suggestions?
 A: I haven't dealt with "closeness" before, but it seems pretty clear what it is from your example. In set-builder notation, I would imagine the definition would be something along the lines of: 
The closeness of a set $A$ to a point $x$ is $$C(A,x) = \inf\{\left|a-x\right|: a\in A \}$$ In "conversational" speak, the above means that for every element $a$ in your set $A$ (we can use the set $\{2,3,5 \}$ that you listed above to make this easier) we need to consider the magnitude of $|a-x|$ where $x$ is the point you need to determine the closeness of with respect to $A$. In your example you have $x=0$, so we will work with that as well. Hence, we need to consider $|2-0|, |3-0|$ and $|5-0|$. The "inf" portion means we apply the infimum operation on our set, which means we want the smallest magnitude of our three choices (for us that is $|2-0|=2$). For the other example set you have above $\{1.31, 1.301, \dots \}$ we must perform a similar process. This set is infinite, so obviously it is a waste of time to calculate infinitely many magnitudes of the form $|1.31-0|, |1.301-0|, \dots$ Instead we can cite the limit of the set and use that to calculate the the closeness to zero. Note that each element can be written as $1.3+\frac{1}{10^n}$ for natural numbers $n \geq 2$. Hence we have a set of strictly decreasing elements $$\{1.31, 1.301, 1.3001 \dots \} = \left \{1.3+\frac{1}{10^2}, 1.3+\frac{1}{10^3}, 1.3+\frac{1}{10^4}, \dots \right \}$$ Again noting the fact that we have a strictly decreasing sequence of numbers, then $$\inf \left\{\left|1.3+\frac{1}{10^n}-0\right| \right \} = \lim_{n \to \infty} \left|1.3+\frac{1}{10^n}-0\right| \\ = |1.3-0| \\ = 1.3$$ Hopefully some of this made sense. Let me know if I am still being confusing and I will try to clarify more!
