Consider the sequence given by the recursion a₀ = 1, a₁=1/3, a_n+1 = 2/3*a_n - a_n-1 (n ≥ 1).  Prove that there exists a positive integer n such that a_n > 0.9999.

I wrote a program, and I found that a₂₄₅ = 0.999969287.  Can anybody give an exact mathematical proof?

Wow, that sequence is a real cement mixer!  It seems to rise and fall like the tide, with a period of about 6411/1256.

I notice a₈₃₂ = 0.999999529702558, and a₆₄₁₁ = 0.999999996898893.

The "high tides" also follow a cycle with a non-integral period of about 6411/131.

I would make a conjecture that for any ε, an n can be found such that 1-a_n < ε.  I would like to see a proof of that.

Here is a graph of a_n vs. n:

In the above graph, you can see that there is a local maximum with a period of about 5, but not exactly 5. Furthermore, there is a second-order period that is apparent in which the local maxima get closer to one about every 50 iterations, but, again, not exactly 50.

Here is a graph of the apparent period of this function vs. n:

And here is a graph of the apparent second-order period that you can see in the first graph.