Convergence of an=(1−12)(12−13)…(1n−1n+1)a_n=(1-\frac12)^{(\frac12-\frac13)^{…^{(\frac{1}{n}-\frac{1}{n+1})}}} [closed]

I’m interesting to see the opinion of MO about my question which I posted here in SE, Answers I received have not convinced me, And no clear proof posted there only numerical computation are provided. I should post the question as it is in order to know more about convergence of the titled sequence

Question:
let us moving to telescopic sum using exponent, Assume we have this sequence: $a_n=(1-\frac12)^{(\frac12-\frac13)^{...^{(\frac{1}{n}-\frac{1}{n+1})}}}$ with $n\geq1$, this sequence can be written as power of sequences: ${x_n} ^ {{{y_n}^{c_n}}^\cdots}$ such that all them value are in $(0,1)$, I want to know if the titled sequence should converge to $1$? and how we can evaluate it for $n$ go to $\infty$?

Answer

Attribution
Source : Link , Question Author : Zeraouli Rafik , Answer Author : Community