What is the meaning of "prove by example"? In page 9 of Mathematical Analysis I by Vladimir A. Zorich, I saw the idiom "prove by example":

As you can see, it is completely different from what is described in this post:
Is there such a thing as proof by example (not counter example)
But when I search in Google, did not find any definition for that. Now I want to know is that a standard idiom? If yes, what is the meaning of that?
 A: I'm not an English native ... but I think that the meaning is simply : 

"We will not prove all the stated theorems but, in order to show the method of proof (by way of example), we will prove one of them."

Similar : for example, by way of illustration.
A: This is more of a general grammatical point.  
Doing something "by way of example" means that you are using it as an example of a typical way of doing something. For example in this case, there are two equalities to be proved, and the two proofs are very similar. Therefore they show you the first proof, and leave the second as an exercise.
A: I use "proof by example" to mean that the fact is demonstrated with an example in such a way that makes the method of the general proof clear. For example, if the proof of a number theoretic statement involves a process depending on the prime factorization, walking through the process with, say, $1500 = 2^2\cdot 3 \cdot 5^3$ would allow a clear demonstration of what to do with primes from each class modulo 4 and with various powers.
I would not have used that phrase for this example, where (as Mark S. noted) it was proof by picking an arbitrary element. 
