Proof series decreases by induction

Question

I have a question regarding a proof by induction. We have to see whether or not the following series converges. $$U_n = \frac{1 \cdot 4 \cdot 7 \cdots (3n - 2)}{2 \cdot 5 \cdot 8 \cdots (3n-1)}$$ I was trying to do this by proving that this series has a lower limit of $0$ and is decreasing. It's easy enough to see that it has a lower limit of $0$ but proving (by induction) that this series is decreasing has proven to be difficult. I understand that I need to prove that $u_n > u_{n+1}$ but I have no idea how to go about doing this.

Answer 1

$$u_n>\frac{3n-2}{3n-1}u_n=u_{n+1}$$ and so by induction the entire sequence is monotonic decreasing.

Answer 2

Let $a_n = \dfrac{1*4* .. *(3n - 2)}{2*5 .. *(3n - 1)}$.

The sequence converges from the Monotonic Convergence Theorem, as it is bounded below (by zero) and is decreasing:

$a_{n + 1} = a_n\dfrac{3n + 1}{3n + 2} < a_n$