Let $l=\{(a_n)_{n \in ℕ}|a_n \in \Bbb R \text{ and } \sup|a_n|<\infty\}$ If $a=(a_n)$ and $b=(b_n)$ are in $l$, define a metric on $l$ by
$d(a,b)=\sup_{n \in \Bbb N}|a_n-b_n|$
Prove that $d$ is a metric:
(1) $d(a,a)=\sup|a_n-a_n|=\sup|0|=0$ for all $a_n in l$
(2) Suppose $d(a,b)=\sup|a_n-b_n|$ Then, $\sup|a_n-b_n|=\sup|-(-a_n+b_n)|=\sup|-1||b_n-a_n|=\sup|b_n-a_n|=d(b,a)$ for all $a_n,b_n$ in $l$.
(3) Let $a,b,c$ be in $l$. Then
$d(a,c)=\sup|a_n-c_n|=\sup|a_n-b_n+b_n-c_n|=\sup|(a_n-b_n)+(b_n-c_n)|\le\sup|a_n-b_n|+\sup|b_n-c_n|=d(a,b)+d(b,c)$
Is this correct?
Thanks