User:Caliburn/s/ra/Arithmetic Mean of Sequence tends to Limit of Sequence

Theorem
Let $x \in \R$.

Let $\sequence {x_n}_{n \in \N}$ be a sequence of real numbers converging to $x$.

Then:


 * $\ds \sequence {\frac 1 n \sum_{i \mathop = 1}^n x_i}_{n \in \N}$ converges

and in particular:


 * $\ds x = \lim_{n \mathop \to \infty} \paren {\frac 1 n \sum_{i = 1}^n x_i}$