Here's Prob. 4, Sec. 1.2 in the book Introductory Functional Analysis With Applications by Erwine Kreyszig:
Find a sequence which converges to $0$, but is not in any space $\ell^p$, where $1 \leq p < +\infty$.
My effort:
For any $n \in \mathbb{N}$, we have the following chain of inequalities. $$ 0 < \sqrt[p]{2n - \sqrt[n]{n}} \leq 2n - \sqrt[n]{n} < 2n;$$ so $$ 0 < \frac{1}{2n} < \frac{1}{2n - \sqrt[n]{n}} \leq \frac{1}{\sqrt[p]{2n - \sqrt[n]{n}} }.$$ Now as the series $\sum \frac{1}{2n}$ diverges to $+\infty$, so does each of the series $\sum \frac{1}{2n - \sqrt[n]{n}} $ and $\sum \frac{1}{\sqrt[p]{2n - \sqrt[n]{n}} }$.
Now we take the sequence $\left( \xi_n \right)_{n \in \mathbb{N}}$, where $$\xi_n \colon= \frac{1}{\sqrt[p]{2n - \sqrt[n]{n}} }$$ for each $n \in \mathbb{N}$. Then we note that $$\lim_{n \to \infty} \xi_n = 0,$$ but this sequence is not in $\ell^p$, for any fixed $p$ such that $1 \leq p < +\infty$.
Am I right? If so, is this example simple enough? Or, can we come up with one which is even simpler? If not, then where am I going wrong?