Let $l_\infty$ be the vector space of bounded sequences of real numbers. Let $\|\mathbf{x}\|=\sup_{n\ge 1}|x_n|$. Prove that $\|\mathbf{x}+\mathbf{y}\|\le\|\mathbf{x}\|+\|\mathbf{y}\|$.
My attempt:
Let $\mathbf{x}, \mathbf{y}\in l_\infty$. Then $\mathbf{x}=(x_1,\ x_2,\ \ldots)$ and $\mathbf{y}=(y_1,\ y_2,\ \ldots)$. $\mathbf{x}+\mathbf{y}=(x_1+y_1,\ x_2+y_2,\ \ldots)$
$\|\mathbf{x}\|=\sup_{n\ge 1}|x_n|\implies |x_n|\le\|\mathbf{x}\|\ \forall n\ge 1$
$\|\mathbf{y}\|=\sup_{n\ge 1}|y_n|\implies |y_n|\le\|\mathbf{y}\|\ \forall n\ge 1$
$\|\mathbf{x}+\mathbf{y}\|=\sup_{n\ge 1}|x_n+y_n|\implies |x_n+y_n|\le\|\mathbf{x}+\mathbf{y}\|\ \forall n\ge 1$
$|x_n+y_n|\le|x_n|+|y_n|\ \forall n\ge 1$ (Triangle inequality)
But $|x_n|\le\|\mathbf{x}\|$ and $|y_n|\le\|\mathbf{y}\|\ \forall n\ge 1$
So, $|x_n+y_n|\le\|\mathbf{x}\|+\|\mathbf{y}\|\ \forall n\ge 1$
So, $\|\mathbf{x}\|+\|\mathbf{y}\|$ is an upper bound for $|x_n+y_n|$. --------------- (1)
So, $\sup_{n\ge 1}|x_n+y_n|\le\|\mathbf{x}\|+\|\mathbf{y}\|$
$\Leftrightarrow\|\mathbf{x}+\mathbf{y}\|\le \|\mathbf{x}\|+\|\mathbf{y}\|$
QED