Is this a true inequality

Question

Suppose that $x_1,x_2,...x_N$ are real vectors with the dimension of $n$. Define $X=[x_1^T,...,x_N^T]^T$. Is it true to say: $||X||\le||x_1||+||x_2||+...+||x_N||$ where $||X||$ is the Euclidian 2-norm of the vector $X$?

Answer 1

\begin{align} X = [x_1^T,...,x_N^T]^T = [x_1^T,0 ,..., 0 ]^T + \ldots + [0,0,...,0,x_N^T]^T \end{align}

By triangle inequality.

\begin{align} \left\|X \right\| &\le \left\|[x_1^T,0 ,..., 0 ]^T\right\| + \ldots + \left\|[0,0,...,0,x_N^T]^T \right\| \\&=\sum_{i=1}^n \left\| x_i\right\| \end{align}