Showing that the sequence $ x_n = \frac {1}{1 + x_{n-1}} $ is convergent

Question

Sequence is recursively defined by $ x_0 = 1 $

I managed to show it is boundness by showing that $ 0 \lt x_n \lt 1 $

Now, when i try to show monotony of the sequence i got the problem because sequence is neither increasing or decreasing. I don't know what to do here. Thanks in advance

Answer 1

Hint: Try to show that $\{x_{2n}\}$ is monotonically increasing while $\{x_{2n + 1}\}$ is monotonically decreasing, and they converge to the same limit.

For a similar problem and the complete answer, check this post.

Answer 2

Hint. Often the easiest way to do this sort of problem is to find the limit if it exists, then prove that that limit actually is correct.

In this case it is clear that the limit, if it exists, is positive and satisfies $$L=\frac1{1+L}\ ,$$ so $L^2+L-1=0$ and hence $$L=\frac{\sqrt5-1}{2}\ .$$ Now we have $$\left|x_n-L\right|=\Bigl|\frac1{1+x_{n-1}}-L\Bigr| =\Bigl|\frac{1-L-Lx_{n-1}}{1+x_{n-1}}\Bigr|\ ;$$ replacing $1-L$ by $L^2$ and doing a bit of algebra, $$\left|x_n-L\right|=\Bigl(\frac{L}{1+x_{n-1}}\Bigr)\left|x_{n-1}-L\right|\ .$$ Since $0<L<1$ and $1+x_{n-1}>1$ we have $\left|x_n-L\right|\to0$ as $n\to\infty$, and this completes the proof.