= Solution
Consider
$$
W=\{w\in B:w(j)\in\{-2^j,2^j\}\text{ for every }j\in\mathbb N_0\}.
$$
The map sending a sequence of signs $(\varepsilon_j)$ to $(\varepsilon_j2^j)$ is a bijection from $\{-1,1\}^{\mathbb N_0}$ onto $W$. This is an <uncountable set> by the <Cantor diagonal argument>: given a proposed list $w_0,w_1,\ldots$ of its elements, define $w(j)=-w_j(j)$. Then $w\in W$ differs from $w_j$ at coordinate $j$ for every $j$.
Since $\ker\pi$ is <countable>, $W$ cannot be contained in it. Consequently there exists
$$
\boxed{w\in W\setminus\ker\pi,\qquad\pi(w)\ne0}.
$$
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