convergence of a tower power

Prove that the sequence of general term $(\frac 12)^{(\frac 13)^{(\frac 14)^{...\frac 1n}}}$ is convergent. the three dots are antidiagonal of course :) My try was to compare it with some easier one like the power tower of $\frac 12$. I conjectured with wolframalpha that it can be divided to two subsequences which seem to be adjacent but i'm finding it hard to prove it Can anyone give some hints.

Here's the result of computing your sequence with Mathematica. It is apparent that odd and even subsequences separately converge to two different limits.

f[n_] := Fold[Power[#2, #1] &, 1, Table[1/k, {k, n, 2, -1}]]

N[f[500], 20]
0.65836559926633118818

N[f[1000], 20]
0.65836559926633118818

N[f[499], 20]
0.69034712611496431947

N[f[999], 20]
0.69034712611496431947

DiscretePlot[f[n], {n, 100}]