Correct solutions were received from Juan Carlos Marivela, Lou Cairoli, Jens Vob, Philippe Fondanaiche.

The stated property is correct.  The easiest proof, I think, proceeds by induction.  I won't post a solution so as not to spoil your fun.  But here's a hint -- it's easier to prove the harder statement

2Ýn < 3Ý(n-1) < 2Ý(n+1)/2.

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