Left Shift Operator is Linear Mapping

Theorem
Let $X = Y = \ell^2$ be 2-sequence spaces over real numbers.

Let $L : X \to Y$ be the left shift operator.

Then $L$ is a linear mapping.

Proof
Let $x = \tuple {x_1, x_2,x_3, \ldots}, y = \tuple {y_1, y_2, y_3, \ldots} \in \ell^2$

Let $\alpha \in \R$.