Here we take$$\mathbb{N} := \{ 1, 2, 3, \ldots \}.$$
Let $\ell^2$ denote the set of all the real (or complex) sequences $\left( \xi_i \right)_{i \in \mathbb{N} }$ such that the series $\sum \left\lvert \xi_i \right\rvert^2$ is convergent. For each $\mathbf{x} := \left( \xi_i \right)_{i \in \mathbb{N}}$, $\mathbf{y} := \left( \eta_i \right)_{i \in \mathbb{N}}$ in $\ell^2$, let$$d( \mathbf{x}, \mathbf{y} ) := \sqrt{ \sum_{i=1}^\infty \left\lvert \xi_i - \eta_i \right\rvert^2 }.$$
Then how to show that, for any non-empty open set $G$ in the metric space $\ell^2$, there exists a point$\mathbb{p} := \left( \rho_i \right)_{i \in \mathbb{N} }$ in $G$ such that $\rho_1 \neq 0$?
My attempt:
Let us assume the contrary. That is, let us assume that, there exists a non-empty open set $G_0$ in the space $\ell^2$ such that, for every point $\mathbf{x} := \left( \xi_i \right)_{i \in \mathbb{N} }$ in $G_0$, we must have $\xi_1 = 0$.
How to proceed from here?
PS: Owing my obligation to the comment below, I think we can proceed as follows:
Let the real number $\epsilon > 0$ be arbitrary. If $\mathbf{x} := \left( \xi_i \right)_{i \in \mathbb{N}}$ be any point in $\ell^2$ such that $d(\mathbf{x}, \mathbf{p} ) < \epsilon$, where $\mathbf{p} := \left( \rho_i \right)_{i \in \mathbb{N} } \in \ell^2$, then we must have$$\left\lvert \xi_1 - \rho_1 \right\rvert \leq \sqrt{ \sum_{i=1}^\infty \left\lvert \xi_i - \rho_i \right\rvert^2} < \epsilon, $$and so we must have$$\left\lvert \xi_1 - \rho_1 \right\rvert < \epsilon. $$On the other hand, if, for each $i = 1, 2, 3, \ldots$, we take$$ \xi_i := \rho_i + \frac{\epsilon}{2^i}, $$for example, then we have$$d( \mathbf{x} , \mathbf{p} ) = \sqrt{\sum_{i=1}^\infty \frac{\epsilon^2}{4^i} } = \frac{\epsilon}{\sqrt{3}} < \epsilon.$$
Thus no open ball in $\ell^2$ can be contained in the set$$S := \left\{ \left( \xi_i \right)_{i \in \mathbb{N}} \in \ell^2 \colon \xi_1 = 0 \right\}. \tag{1} $$But since every non-empty open set in $\ell^2$ is a union of open balls, no non-empty open set in $\ell^2$ can be contained in the set $S$ in (1).
Is this proof satisfactory enough?