Prove that a stable algorithm cannot solve an ill-conditioned problem with a precision greater than that of input values
Let's try to define those terms in math language:


Denote $x$ as the input, and $f(x)$ is the solution to the problem with input $x$.
Ill-condition problem: $x'=x + \delta \Rightarrow f(x') \gg f(x) + \delta$
Stable algorithm $F(x) \simeq f(x')$ (stable algorithm give you result close to result of a nearby input)


So apply stable algorithm on ill-condition problem we get: $F(x) \simeq f(x') \gg f(x) + \delta$

Which says the best you can get can't be smaller than error in the input.

I think you can also prove by contradiction, saying if the stable algorithm can do that, then the changes in input will make only small change in the output which makes the problem well-condition.