1. **Stating the problem:** We want to verify the inequality and the recurrence relation given: $$U_{n+1} < \frac{2}{5} U_n \quad \forall n$$ 2. **Given expressions:** The user manipulates expressions involving $U_n$ and $U_{n+1}$ and arrives at the inequality: $$\frac{U_{n+1}}{\frac{2}{3} U_n} \leq 0$$ which simplifies to $$U_{n+1} < \frac{2}{5} U_n$$ 3. **Checking the steps:** - The user starts with expressions like $$\frac{2 U_n}{2 U_n / 5}$$ and simplifies terms involving $U_n$. - The key step is the inequality: $$10 U_n - 4 U_n + 10 U_n - 5 (2 U_n + 5) \leq 0$$ which simplifies to $$-4 U_n \leq 0$$ 4. **Interpretation:** Since $-4 U_n \leq 0$ implies $U_n \geq 0$, the inequality holds if $U_n$ is non-negative. 5. **Conclusion:** The final inequality $$U_{n+1} < \frac{2}{5} U_n$$ holds under the assumption that $U_n \geq 0$. Therefore, the steps and inequality are correct given the assumptions. **Final answer:** The inequality $$U_{n+1} < \frac{2}{5} U_n$$ is correct for all $n$ assuming $U_n \geq 0$.