Left Shift Operator on 2-Sequence Space is Continuous

Theorem
Let $\struct {\ell^2, \norm {\, \cdot \,}_2}$ be the $2$-sequence space with $2$-norm.

Let $L : \ell^2 \to \ell^2$ be the left shift operator.

Then $L$ is continuous on $\struct {\ell^2, \norm {\, \cdot \,}_2}$.

Proof
Let $\sequence {a_n}_{n \mathop \in \N} = \tuple {a_1, a_2, a_3, \ldots}$ be a $2$-sequence.

By Continuity of Linear Transformation between Normed Vector Spaces, $L$ is continuous in $\struct {\ell^2, \norm {\, \cdot \,}_2}$.