Positive Sub-Martingale with Limit = 0?

Question

We have $u(t+1)=u(t)r_{t+1}$, with $u(0) = 1$, and $r_t$ is a random variable with 2 possible values:

$2/9$ appears with the possibility of 80%,

$8$ appears with the possibility of 20%.

Then we have $$ \lim_{t\to\infty}E[u(t)]=\lim_{t\to\infty}\left[\left(\frac{2}{9}\right)^{0.8t}\times (8)^{0.2t}\right] = \lim\limits_{t\to\infty}(0.300\times1.516)^t = 0 \dots\dots(1) $$

$$ \lim_{t\to\infty}E[u(t)]= \lim_{t\to\infty}\left(E[r_1]E[r_2]\dots E[r_t]\right)=(\frac{2}{9}\times0.8+8\times0.2)^t=\infty. \dots\dots(2) $$

Question 1: (1) and (2) seems to conflict with each other.

At the same time, we have

\begin{align}\mathbb{E}[u(t+1)|\mathcal{F}_t]-u(t)&=u(t)(\mathbb{E}[r_{t+1}|\mathcal{F}_t]-1) \\ &=u(t)\left(0.8\times\frac{2}{9}+0.2\times8 - 1\right) \\ &= 0.778u(t)>0. \end{align}

It seems that $u(t)$ is a sub-martingale.

Question 2: $u(t)$ is a sub-martingale seems to conflict with (1).

import numpy as np
import random as random

u = 1
n = 100

for n in range(n):
    r = random.random()
    if r < 0.8:
        u = u * 2 / 9
    else:
        u = u * 8

print(u) # the outputs are always like 2.9307827938878015e-44

Question 3: Can we say $u(t)$ converge in distribution/probability to zero, how to prove it?

I'm a beginner with stupid questions, these questions mess me up, don't know where are the mistakes or misunderstandings.