How this inequality is derived? Let $T$ ∶ $ℓ_2$ → $ℓ_2$ be defined by
$T((x_1,x_2,...,x_n...))$=$(X_2-X_1, X_3-X_2,...,X_{n+1}-x_n,...)$
Then I have find the norm of $T$. 
Here is the answer to this question:  https://math.stackexchange.com/a/1647794/581242
I am not able to see how this first inequality is derived.
$\|(Tx)\| = \sqrt{\sum_{i=1}^\infty |x_{i+1}-x_i|^2} \leq 
\sqrt{\sum_{i=1}^\infty |x_{i+1}|^2 + \sum_{i=1}^\infty|x_i|^2} \leq 2\|x\|$

It appears that there was a small typo. Use the inequality $|a-b|^{2} \leq 2(a^{2}+b^{2})$ . The final result is correct but the first inequality is wrongly stated. 
