Left shift on bounded sequences
= Left shift on bounded sequences
{title2=$S(x_1,x_2,\ldots)=(x_2,x_3,\ldots)$}
The left shift is a <bounded linear operator> of <operator norm> one on the <l-infinity sequence space>. For $y=x-Sx$, telescoping gives $N^{-1}\sum_{n=1}^Ny_n=(x_1-x_{N+1})/N\to0$. This identity is useful in constructing a <Banach limit>.