Let a constant $B \ge 1$, and let $l_1 = 0$, $b_1 = 0$ be the values of $l$ and $b$ (respectively) at time $t = 1$.

Let $l_{t+1} = l_t + 1$ if $b_i < B$, and $l_{t+1} = l_t$ otherwise

Let $b_{t+1} = b_t + \frac{l_{t+1}}{t}$

So here I can intuitively see that $\forall t \le B: l_t = b_t = t$, and that $\forall t > B: l_t = B$. So $\forall t: l_t \le B$ (i.e. $l$ is bounded above by $B$).

I have two questions:

Can we also give an upper bound on $l_t$ if it was defined as: Let $l_{t+1} = l_t + 1$ if $b_i < B$ and with probability $p_t$, and $l_{t+1} = l_t$ otherwise. That is with a probability $p_t$ the value of l is increased if $b_i < B$.

Note: Here is the simple algorithm corresponding to these functions to better explain the problem.

```
let b=0, l=0, T=0.7, N=100
for t from 1 to N:
get the observation x_t
let P(x_t) the probability associated with the random variable x_t
if P(x_t) > T and b < B then l := l + 1
b := b + ( l / t )
```