# Are 2 norm and infinity norm equivalent in an infinite dimensional vector space space

I am trying to answer the following question:

Are the 2-norm and the infinity norm equivalent in $$l^2$$, the space of real valued sequences which are square summable?

($$l^2 := \big\{ x = (x_i)_{i \in \mathbb{N}} \ \ \big| \ \ \sum_{i=1}^{\infty} |x_i|^2 < +\infty \big\}$$)

I am aware that in an n-dimensional vector space, we are able to obtain the following inequality (whose lower bound still stands for an infinite dimensional space):

$$\|x\|_\infty \le \|x\|_2 \le \sqrt{n} \|x\|_\infty$$

But I am unsure of how to proceed. Are there any obvious sequences to use as counterexamples if the norms are not equivalent? Otherwise can someone give me a hint for how I can find a constant to replace $$\sqrt{n}$$ in the upper bound? Thanks!

Take $$x_n = (1,1,1,...,\underset{n-th}{1}, 0,0,....)\in \ell^2.$$ Then $$||x_n ||_{\infty} =1$$ but $$||x_n||_2 =\sqrt{n}$$ so these norms can't be equivalent on $$\ell^2$$