I encountered this theorem on Wiki
"there does not exist an injective function whose codomain is smaller than its domain".
Here is my attempt to prove it, can you please have a look if this proof is OK?
Proof:
Let's say we have:
$f:A\to B$
where $|A| = N$ and $|B| = M$.
Let's take all elements from $A$:
$ a_1,\cdots ,a_N $
and by mapping them, we get:
$f(a_1),\cdots , f(a_N)$.
There can't exist two same elements in latter list (according to injection).
Which means there need to be at least N distinct elements in B. This can't hold if $M < N$.
Best Answer
The continuous case is not true, $f(x) = x/2$ for $0\le x\le 1$, so you really need the assumption that the domain is a finite set.
Given a finite set as domain, I think your proof is OK.