Let $X,Y$ be Banach spaces and $\phi_n\in\mathcal{L}(X,Y)\backslash\{0\}$. Show $\{ x\in X: \phi_n(x)\ne 0,\forall n\in \mathbb{N}\}$ is dense in $X$

Let $X,Y$ be Banach spaces and $\phi_n\in\mathcal{L}(X,Y)\backslash\{0\}$. Show $\{ x\in X: \phi_n(x)\ne 0,\forall n\in \mathbb{N}\}$ is dense in $X$.

I want to show this using Baire's theorem. Let $U_n :=\{ x\in X: \phi_n(x)\ne 0\}$, then it suffices to show that $U_n$ is open and dense in $X$.

Let $x \in U_n$, then $\phi_n(x)\ne 0$ in $Y$. Because $\phi_n$ is continuous, there exists an open neighborhood $W$ of $x$ in $X$ such that $\forall w\in W: \phi_n(w)\ne 0$. This shows that $x$ is an interior point of $U_n$, hence $U_n$ is open in $X$.

Let $x\in X$ and suppose that $\exists W\in\tau_x, x\in W: W\cap U_n = \emptyset$. This implies $\phi_n(x)=0$. How do I proceed from here? I haven't used $\phi_n\ne 0$ yet, so I feel like this would be the contradiction we're looking for.

Claim. if a linear map $L$ restricted on a ball $B(x,r)$ is completely zero, then $L$ is completely zero.

To see this, fix the $x,r$. For every $y\in X$ we have

$$L(B(y,r)) = L(B(x,r) + y - x)$$

Hence the $L$ restricted on $B(y,r)$ is constant. This forces the $L$ is zero. (you can expand $B(x,r)$ to the whole space $X$)