# Infinite Limit Operator is Linear Mapping

## Theorem

Let $c$ be the space of convergent sequences.

Let $\R$ be the set of real numbers.

Let $L : c \to \R$ be the infinite limit operator.

Then $L$ is a linear mapping.

## Proof

Let $\mathbf x = \sequence {x_n}_{n \mathop \in \N}, \mathbf y = \sequence {y_n}_{n \mathop \in \N} \in c$.

Suppose $\mathbf x$ and $\mathbf y$ converge to $x$ and $y$ respectively.

Let $\alpha \in \R$.

### Distributivity

 $\ds \map L {\mathbf x + \mathbf y}$ $=$ $\ds \map {\lim_{n \mathop \to \infty} } {x_n + y_n}$ Definition of Infinite Limit Operator $\ds$ $=$ $\ds x + y$ Sum Rule $\ds$ $=$ $\ds \lim_{n \mathop \to \infty} x_n + \lim_{n \mathop \to \infty} y_n$ $\ds$ $=$ $\ds \map L {\mathbf x} + \map L {\mathbf y}$ Definition of Infinite Limit Operator

$\Box$

### Positive homogenity

 $\ds \map L {\alpha \mathbf x}$ $=$ $\ds \map {\lim_{n \mathop \to \infty} } {\alpha x_n}$ $\ds$ $=$ $\ds \alpha x$ Multiple Rule $\ds$ $=$ $\ds \alpha \lim_{n \mathop \to \infty} {x_n}$ $\ds$ $=$ $\ds \alpha \map L {\mathbf x}$ Definition of Infinite Limit Operator

$\blacksquare$