If $1 ≤ p < q < \infty$, then the identity operator $\Lambda:l^p → l^q $ defined by $Λx = x$ is continuous.
Is this a right way to write a proof of this statement?
Taking norm from both sides of $\|\Lambda x\|_{l^q} = \|x\|_{l^q} \le \|x\|_{l^p}$ with $1 ≤ p < q < \infty$ . Then as $x \in l^p$ , by the inclusion $l^p \subset l^q$ , then the operator is bounded and hence continuous.