Proof of Markov propoerty for Ehrenfest urn

In many books Ehrenfest Urn is used as an example of a homogeneous Markov chain, where entries in transition probabilities depend on the state, e.g.

$$p_{i,i+1} = \frac{i}{n}, q_{i,i-1} = 1- \frac{i}{n}$$

or, in recurrent notation, for period $t$:

$$S_{t} = S_{t-1} + \xi_{t-1}$$ where $\xi_{t-1}$ is a random variable. What I have never seen is the proof that this process satisfies weak Markov property:

$$P(S_{t+1} = i_{t+1}|S_{t} = i_{t},...,S_{0}=i_{0}) = P(S_{t+1} = i_{t+1}|S_{t} = i_{t})$$

Only in Flajolet, Dumas,Puyhaubert(2006) on p.70 it is mentioned that Ehrenfest urn can be viewed as a random walk on N-dimensional cube, but I can't relate it.

I'd massively appreciate suggestion on how to approach this proof.

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This question is a totally fair question, but doesn't belong on this site - try asking it on <a href="math.stackexchange.com">math.stackexchang…; where you should get some good answers. – Anthony Quas Feb 26 2012 at 1:24
OK, I'll wait for some time. If others agree, I'll move this question to MSE. – sigma_z_1980 Feb 26 2012 at 1:27