Show that the $ℓ^1$ and $ℓ^2$ norms are not equivalent on $ℓ^1$ Show that the $ℓ^1$ and $ℓ^2$ norms are not equivalent on $ℓ^1$.
I am given a hint of Find elements $x_n ∈ ℓ^1$ such that $||x^n||_{ℓ^2} ≤ M$ for some $M$ but $||x_n||_{ℓ^1} → ∞ $as $n → ∞$. but I am unsure on how to tackle this question. Any help greatly appreciated!

Hint: Let $$x_n = \left(1,\frac{1}{2},\dots,\frac{1}{n},0,0,\dots\right).$$
Edit:
$$\|x_n\|_{\ell_1} = \sum_{k=1}^n\frac{1}{k}\to\infty$$
$$\|x_n\|_{\ell_2}^2 = \sum_{k=1}^n\frac{1}{k^2}\to\frac{\pi^2}{6}$$
