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Let ${u_n}$ be a real sequence defined by $ u_1=2, ~ u_2=0, ~ u_{n+2 } =\frac{1}{2^{u_n}} + \frac{1}{2} $ Prove that ${u_n}$ has finite limit and find $\lim u_n$.

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    This series is composed of 4 sub series, all of which are monotonic and bounded, try to show that, and show that all converge to the same limit.2012-11-26

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HINT: Note first that the subsequences $\langle x_{2k}:k\in\Bbb Z^+\rangle$ and $\langle x_{2k-1}:k\in\Bbb Z^+\rangle$ are completely independent, so you’re really dealing with the recurrence $y_{n+1}=\frac1{2^{y_n}}+\frac12\tag{1}$ with two different initial values, $2$ and $0$. You need to show that the resulting sequence converges for each of these initial values, and to the same limit for both.

Let $f(x)=\frac1{2^x}+\frac12\;,$ and let $g(x)=f(x)-x$. Show that the limit of a convergent sequence satisfying $(1)$ must be a fixed point of $f$ and therefore a zero of $g$. Then show that $g'(x)<0$ for all $x$ and conclude that $g$ has only one zero. By inspection the unique fixed point of $f$ is $x=1$.

Finally, show that if $x<1$, then $x, and if $x>1$, then $x>f\big(f(x)\big)>1$, and put the pieces together to get the desired result.