"Assume that $y=(b_{n})_{n \geq 1} \in \ell^{3}$. For each $n\geq 1$, define the vector
$$Y_{n} = (b_{n+1},b_{n+2},\ldots ) = (b_{n+k})_{k \geq 1}.$$
Prove that the sequence of vectors $(Y_{n})_{n=1}^{\infty}$ converges to 0 in $(\ell^{3},\| \cdot \|_{3})$."
I'm having a lot of trouble with this question. I know that for $(Y_{n})_{n\geq 1}$ to converge to $0$ we must have that $\| Y_{n} - 0 \|_{3} = \|(Y_{n})_{n\geq 1} \|_{3} \to 0$ as $n \to 0$. So we must have
$$\left(\sum_{n=1}^{\infty} \mid Y_{n} \mid ^{3}\right)^{1/3} = \left (\sum_{n=1}^{\infty} \mid b_{n+k} \mid ^{3}\right)^{1/3} \to 0$$
However, im having a lot of trouble showing this. I thought about changing the starting index to be $n+k$ but that didn't seem to give me any other route forward. Any help would be appreciated.