If $U_0 = 0$ and $U_n=\sqrt{U_{n-1}+(1/2)^{n-1}}$, then $U_n < U_{n-1}+(1/2)^n$ for $n > 2$ Letting $$U_n=\sqrt{U_{n-1}+(1/2)^{n-1}}$$ where $U_0=0$, prove that:
$$U_n < U_{n-1}+(1/2)^n$$ where $n>2$

Here is a useful factoid:

For every $x\geqslant1$ and $y\gt0$, $\sqrt{x+2y}\lt x+y$.

Now, apply this to your setting. First note that $U_n\geqslant1$ implies $U_{n+1}\geqslant1$. Since $U_1=1$, this proves that $U_n\geqslant1$ for every $n\geqslant1$. Then, choosing $n\geqslant2$, $x=U_{n-1}$ and $y=1/2^n$, the factoid yields $U_n\lt U_{n-1}+1/2^n$, as desired.
Finally the result holds for every $n\geqslant2$ (and not only $n\gt2$).
Can you prove the factoid above?
