Convergent Real Sequence/Examples/x n = root x n-1 y n-1, 1 over y n = half (1 over x n + 1 over y n-1)

Example of Convergent Real Sequence
Let $\sequence {x_n}$ and $\sequence {y_n}$ be the real sequences defined as:


 * $x_n = \begin {cases} \dfrac 1 2 & : n = 1 \\ \sqrt {x_{n - 1} y_{n - 1} } & : n > 1 \end {cases}$


 * $\dfrac 1 {y_n} = \begin {cases} 1 & : n = 1 \\ \dfrac 1 2 \paren {\dfrac 1 {x_n} + \dfrac 1 {y_{n - 1} } } & : n > 1 \end {cases}$

Then both $\sequence {x_n}$ and $\sequence {y_n}$ converge to the limit $\dfrac \pi 4$.

Proof
By definition, we have that:
 * $\sqrt {x_{n - 1} y_{n - 1} }$ is the geometric mean of $x_{n - 1}$ and $y_{n - 1}$


 * $\dfrac 1 2 \paren {\dfrac 1 {x_n} + \dfrac 1 {y_{n - 1} } }$ is the reciprocal of the harmonic mean of $x_n$ and $y_{n - 1}$.