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**bkarpuz** First, we show that its decreasing, next we show that it is bounded from below (in worst case it is bounded below by $\displaystyle 0$), this means that it has a limit at infinity.

Then in the recursion we pass to limits on both sides (note that if $\displaystyle x(n)\to\ell$ then $\displaystyle x(n+1)\to\ell$ as $\displaystyle n\to\infty$; that, is every subsequence converges to the same limit).

Finally, we prove that $\displaystyle \ell=1$, which is the desired $\displaystyle \inf$ value since $\displaystyle x$ is decreasing.

I know I repeat the same, but no other idea to show $\displaystyle \ell=1$.