## Limit of binomial supermartingale

Hi!
My first post here, so I'm not sure if this is an appropriate forum for my question. Anyway, here it goes. I am faced with the following problem.

Let X_0 = N and

X_n | X_{n-1} ~ Bin(N, p_n)

with

p_n = X_{n-1}/N*(1-eps(n))

and 1 > eps(n) --> 0. That is, X_n is drawn from a binomial with success probability given by the previous value of the process, and some decreasing sequence eps(n). Clearly, is eps(n) == 0, then X_n == N. On the contrary, if eps(n) is sufficiently large, we expect X_n --> 0.

It is easy to see that X_n is a nonnegative supermartingale and hence converges a.s. to some limit X. It is also easy to see that P(X=0 or X=N) = 1.

However, I want to find minimal conditions on the sequence eps(n) so that X_n --> 0 a.s. My conjecture is that \sum eps(n) = \infty should be enough, but I am not sure how to prove it. Are there any good tricks to identify the limit of a supermartingale?

Regards,
Fredrik