a strange question related to topology 
Give an example or a proof (from whatever branch of mathematics) where
  a desired thing is found by assuming tentatively its existence and
  thereby obtaining clue to its construction.

This appears a completely strange question to me and I wonder exactly what branch of mathematics could contain its proof. I got this question from a textbook of topology and how topology is related to the solution of this, I can't understand. Any help will be appreciated.
 A: Searching the web I found this historical claim copied verbatim in several places,

... so Fourier's contribution was mainly the bold claim 
  that an arbitrary function could be represented by a Fourier series. 

concerning his work in 1807.
His work was reviewed by both Lagrange and Laplace in 1808; they objected to this central thesis. A few years later he was awarded a prize for these contributions, but the report contained this criticism

... the manner in which the author arrives at these equations is not exempt of difficulties and that his analysis to integrate them still leaves something to be desired on the score of generality and even rigour.

See this biography on Jean Baptiste Joseph Fourier, where it states, in regard to Fourier's writings

All these are written with such exemplary clarity - from a logical as opposed to calligraphic point of view - that their inability to persuade Laplace and Lagrange ... provides a good index of the originality of Fourier's views.

Indeed, Fourier was delineating his groundbreaking theory using the mathematical style/logic of that time.
This bring us to the paper
How did Cantor Discover Set Theory and Topology?
S M Srivastava
The opening paragraph:

In order to solve a precise problem on trigonometric series, “Can a function have more than one representation by a trigonometric series?”, the great German mathematician Georg Cantorc created set theory and laid the foundations of the theory of real numbers. This had a profound impact on mathematics. In this article, we narrate this fascinating story

Cantor produced his landmark results on Fourier series in 1870 -1871. 

Fourier's discovery forced a change in mathematical logic and abstraction so that his theories could be made rigorous and precise. The amazing thing is how general set theory became a fundamental cornerstone of mathematics.

See also the video
HISTORY OF FOURIER SERIES AND FOURIER TRANSFORM| Signals and Systems
A: Tychonov's theorem implies existence of choice functions.
