Constructive proofs -- reference Math contests often have questions like

Prove that there are infinitely many numbers such that (insert property).

And the answers to these questions often star with a specific construction, along the lines of

Consider the construction/number/integer....

However the intuition/process behind that construction is more often than not omitted. For some it may just come naturally like a stroke of genius, but I like to think that there is a process of investigation and exploration before the construction presents itself. On these lines, are there any books/articles that discuss the methodology behind constructive proofs through examples?
 A: Here are a couple books that may be useful.
https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995
https://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/069111966X
A: Evening, for starting in proof construction one should think in terms of propositional logic. To comply with the statement above:

Prove that there are infinitely many numbers such that (insert property).

One should first prove there exists at least one element that satisfies the desired property, hence the "Consider the..." hint given. For that construction one should prove that the chosen element has the property as a consequence of other properties of the same element. 
Then one should prove that there are infinitely many elements that satisfy the property. The most common way to do this is suppose that the quantity of elements that satisfy the property is finite, and show that this statement leads to a contradiction (Which is often the existence of another element that satisfies the condition). For example:

We denote $S$ as the set of all numbers that satisfy $A$(the desired property). Consider the element $a$, from Theorem $\Gamma$ we can show $a$ is an element of $S$, hence $a$ satisfies $A$. To show that the set $S$ is infinite we proceed by reductio ad absurdum. We suppose $S$ is finite and contains only the elements $a_1, a_2, \ldots, a_n$. (This is an example of this step) We know $a$ is in $S$, and $a$ has property $B$ which gives us an element $b$ that is in $S$ and can be proved is not any of $a, a_1, a_2, \ldots, a_n$, then there is another element in $S$, which leads us to a contradiction with the statement that $S$ is finite. We can then conclude then that $S$ is finite and that completes the proof.

I, beforehand, apologize for any grammar mistake as English is not first language, but I hope this will satisfy your inquiry.
