I seem to only be able to prove this when I sum from t=0 to t=p. However, when looking at the timeline given d(p)/p, it seems that one should sum from t=0 to t=p-1, which does not seem to lead to the right final answer.
I don't quite see where you are summing to and from with t, maybe change your question. However, that result should hold intuitively if we divide up the interest rate into smaller basic time units. Otherwise an arbitrage opportunity exists!