Consider $(S_n)_n$ a simple random walk, $S_n = \sum_k^n X_k$ where $(X_k)_k$ are i.i.d. random variables with $P(X_k = 1) = P(X_k=-1) = \frac 1 2.$ We further set $X_0=1$ and stop the random walk once it reaches $0$, i.e. $S_n=0$ for some $n\in\mathbb N$. Also, define an excursion from $l$ to $l+1$ of such a random walk as a part of the process where $S_m = l$ and $S_n = l+1$ for some $m<n$ and $l\in\mathbb Z$.
I'm trying to prove the following statement:
Let $l,K\in\mathbb N, l < K$. The expected number of excursions from $l$ to $l+1$ of such a simple random walk conditioned not to hit K, is approximately $$ \left[ 1-\frac{l+1}K \right]^2. $$
So far, I have not made any significant progress. Thus, I'd be glad about any kind of help!