(Continued fraction). Given a positive number a > 0, define inductively the sequence (xn) by

x$_1$ =$\frac1a$ x$_{n+1}$ = $\frac{1}{a+x_n}$

Consider c to be the unique positive root of x$^2$ + ax − 1 = 0.

Show x$_2$ < x$_4$ < · · · < c < · · · x$_{2n−1}$ < · · · x$_3$ < x$_1$, and that lim x$_n$ = c.

So this is a sample problem for my upcoming exam and I'm really stuck on it. I attempted to use the quadratic formula to find c and I know the lim x$_{n+1}$= $\frac{1}{a+c}$ assuming the lim x$_n$=c and then I attempted to use the definition of convergence on x$_{n+1}$ to see if I could get somewhere but I'm lost to be honest. Any tips would be appreciated.


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