I am trying to understand the $l^p$ space and what it means for a sequence to converge in it. I am making an example of my own to see how the norm-converges works, so it maybe wrong etc.
let $p\geq2 $ and take $x_n=(ξ_{nk})_{k\in \mathbb{N}}=(\frac{1}{n},\frac{1}{n^2},\frac{1}{n^3},..., \frac{1}{n^k},...)$
then $|| x_n||=\bigg(\sum^{\infty}_{k=1}|ξ_{nk}|^p \bigg)^{1/p}=\bigg(\zeta (kp) \bigg)^{1/p}=\bigg(\frac{1}{n^p}+\frac{1}{n^{2p}} +\frac{1}{n^{3p}}+... \bigg)^{1/p}$
Letting $n\rightarrow \infty$, $|| x_n|| \rightarrow0$ meanig $||x_n-x||<ε $ where $x=(0,0,...)$ is my reasoning correct ? My professor doesn't explain much, I am a bit confused. If I am wrong, can someone explain how to think about this.