Let $X$ be a normal topological space. Suppose we have a real vector bundle $p: E \to X$ and a point $x \in X$. Let $s: X \to E$ be a section such that $s(x)=0$. I want to show that $s$ can be written as $s=\sum f_is_i$, where $s_i$ are some sections and $f_i$ are continuous functions on $X$, such that $f_i(x)=0$.
My approach goes as follows: Since $p:E \to X$ is a vector bundle we obtain a covering $\{U_i\}$ and a trivialization $p^{-1}(U_j) = U_j \times \mathbb{R}^k$ for some $j$. Then, having a section $s:X \to E$, restricting ourselves to $U_j$, we obtain a function $U_j \to U_j \times \mathbb{R}^k$. We can take the basis of $\mathbb{R}^k$ and thus our section is given now by $k$ real-valued functions on $U_j$. So, the sections we want to obtain would correspond to the basis vectors.
However this approach works only locally and I think the Tietze theorem won't help here.
I would be grateful for a neat argument.