Prove $ \left \| \left [ G \: \: \: \: I \right ] \right \|^{2}_{\infty }\leqslant \left \| G\right \|^{2}_{\infty } +1 $

Question

Consider the Strickly proper transfer function of G(s) and impulse response G(t), by assuming the input signal u(t) having the bounded $\left \| \right \|_{1}$ , How can one prove the following statement: $$ \left \| \left [ G \: \: \: \: I \right ] \right \|^{2}_{\infty }\leqslant \left \| G\right \|^{2}_{\infty } +1 $$