what you've just said says nothing about the length of the hashes, which was my point. You would have to say something like "all possible inputs of a certain length are your pigeons, and these same possibilities(1) are the holes"
(1)this time interpreted as the hash of one and only one of the same list of possible inputs
The proof is not so straightforward, because it's not always true when you expect it to be true. For example, can you hash, reversibly and uniquely, all the numbers between 0 and 10 (real) to all the numbers between 0 and 1?
You would expect the answer to be "no" since there are "more real numbers from 0 to 10 than from 0 to 1". But that's actually not necessarily true the way you might expect, and the answer is actually "yes you can". Simply geometrically establish a coordinate plane like this
a
0 b 1
0 10
so that point (a) has a well-defined coordinate, say (0,1) the 0 from the second line is at the origin, the 1 at the second line is at (1,0) and the bottom 0 is at (0,-1) with the bottom 10 being wherever the ray passing through a and passing through (1,0) - enough to uniquely identify the ray - has the y-value of -1.
Now to get the hash simply move 'b' to wherever between 0 and 1 you're trying to hash, solve for the ray that passes through a and that point, and solve for the x-value of the ray where it has y-value of -1.
Through elementary proof which should be obvious, you'll see that any b casts a distinct ray (which is easily reversible introducing a c into line 3), and you can solve for it if you like.
I'm not saying that you're wrong, but I think the proof isn't as "obvious" and straightforward as you're saying. It doesn't apply to just anything, but to the specific case of the impossibility of hashing distinct length inputs uniquely into a discrete space defined by a uniformly shorter length of bits.
From wikipedia, "A hash function is any algorithm or subroutine that maps large data sets of variable length, called keys, to smaller data sets of a fixed length."
Given the pigeonhole principle, and the observation that there are fewer strings of length K than strings of any length, you have that you cannot map every string into a unique hash. That's what I meant by "a straightforward application of the pigeonhole principle".
If you're dealing with an infinite set of hashes (I'm not sure what that would look like, but hey), then of course you need to take into account the limitations of the pigeonhole principle when dealing with infinite sets - specifically, that it only applies if you have a set of pigeons of a larger cardinality than your set of holes. [0,1] and [0,10] have the same cardinality.
