Convergent Real Sequence/Examples/Arithmetic Mean of Previous 2 Terms

Example of Convergent Real Sequence
Let $\sequence {x_n}_{n \mathop \in \N_{>0} }$ be the sequence in $\R$ defined as:
 * $x_n = \begin {cases} a & : n = 1 \\ b & : n = 2 \\ \dfrac {x_{n - 1} + x_{n - 2} } 2 & : n > 2 \end {cases}$

That is, beyond the first $2$ terms, each term is the arithmetic mean of the previous $2$ terms.

Then $\sequence {x_n}$ converges.

Proof
Let $n > m$.

Then:

Let $\epsilon \in \R_{>0}$ be given.

Let $N$ be sufficiently large that:


 * $\dfrac 1 {2^{N - 2} } \size {b - a} < \epsilon$

Then:


 * $\forall n > N, m > N: \size {x_n - x_m} \le \dfrac 1 {2^{N - 2} } \size {b - a} < \epsilon$

Hence it is seen that $\sequence {x_n}$ is a Cauchy sequence.

Hence the result, by Cauchy Sequence Converges on Real Number Line.