On Convergence of a Sequence

[Pe Joseph-AJP070]

Can anyone help with this problem?
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Define the sequence a(n) by: a(1) = 1; a(n) = 1-(p(n-1)/p(n))*a(n-1) if n >
1, where p(n) denotes the n-th prime. 
It's easy to show (an exercise!) that if L = lim a(n) exists, then L = 1/2. 
Can you prove the convergence of a(n) or the divergence of a(n)? 
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I graphed a(n) from n = 1 to 10^5, and it doesn't look like 2 is a limit,
but at most a cluster point. However,
other people have found "convergence proofs" (all of them erroneous, I
think).

J. L. Pe