I have come across the term “bijective proof” in many numerical analysis papers, but I do not really know its mathematical meaning in, for example, the question on MO asking for an (approximately) bijective proof that $ \zeta(2) = \frac {\pi ^2} 6 $.
What is a “bijective proof” in mathematics and how I can construct one?
In its simplest form a bijective proof means you find a bijection between two finite sets and conclude they have the same number of elements. Sometimes you'll have two collections of sets with each set mapping bijectively into a set of the other class and to show each member of the collection is the same size as exactly one other member of the other collection but the idea is essentially the same.