Finding the norm of the shift operator on $l^\infty$, $(x_1,x_2, \dots)\mapsto (x_2,x_3,\dots)$

$||T||=\sup_{||x||=1} ||Tx||$.

Since $||x||=1\implies \sup \{|x_1|,|x_2|,\ldots |x_n|,\ldots\}=1$.

Note that $A\subset B\implies \sup A\le \sup B$.

So $||Tx||=\sup \{|x_2|,|x_3|,\ldots |x_n|,\ldots\}\le \sup \{|x_1|,|x_2|,\ldots |x_n|,\ldots\}=1\implies ||T||\le1$

Now ,Take $x_0=(1,1,1,\ldots,1)$ , then $Tx_0=(1,1,1,\ldots,1)\implies ||T||\ge 1 $

Combining $||T||=1$.

Clearly $\|Tx\| \le \|x\|$, hence $\|T\| \le 1$. Since $T(0,1,0,...) = (1,0,...)$ we have $\|T\| = 1$.

To show that $\|T\|\geq c$ it suffices to find one $x$ such that $\|Tx\|\geq c\|x\|$, by the definition of the operator norm.

In particular, in your example, to show that $\|T\|\geq 1$ it suffices to find one $x$ such that $\|Tx\|=1$. It is easy to find such examples.