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 $\alpha \in \R$.