Let $w$ be a positive rational number and $w^2>2$. Prove that there exists a positive rational number $x$ such that $x^2>2$ and $x<w$. A condition is you can not use the property of real numbers such as the density property of real numbers (don't say that $w$ is a real number).
The usual answer is to use Newton's method, which preserves rational numbers: $$ x=\frac12\left(w+\frac2w\right) $$
Rudin's book contains a slightly different expression. See here.
You don't really need to do anything fancy here; continuity of the squaring function will do the trick. Consider $$w-\epsilon$$ as $\epsilon \to 0^+$, with $\epsilon>0$ rational (if you like, take a sequence $\epsilon_1, \epsilon_2, \ldots$ of rational numbers decreasing to zero).
Then $(w-\epsilon)^2$ approaches $w^2>2$. Pick $\epsilon$ small enough so that $(w-\epsilon)^2>2$. No reference to real numbers necessary, even though I used continuity.