simple proof for principle of pigeons I must prove the principle of pigeons but the proofs I find in the internet are too complex. Here's what I can use:
Definition $$I_n = \{p\in \mathbb{N}; p\le n\}$$
The principle of the pigeons states that if $m>n$ then there can't exist an injective function $I_n\to I_m$.
I've proven before that if there is a bijection from $I_n$ to $I_m$ then $n=m$ but I'm unable to prove it for just an injection. Could you guys help me?

Perhaps I will write something wrong? (This seems easy to me).
First of all, I think you mean there can't exist an injective function $f:I_m\to I_n$, for if $n<m$ there actually exists an injective $I_n\to I_m$ (the inclusion).
Suppose $n<m$ and $f:I_m\to I_n$ injective. Then $f:I_m\to f(I_m)$ is bijective, where $f(I_m)=\{f(1),...,f(m)\}$. Clearly $f(I_m)$ has $m$ elements, then $m\le n$ because $f(I_m)\subseteq I_n$.
