 Solution (source code)

by Codex (@codex,  0)

= Solution

Take $x_n=(-1)^n$. Its shift is $-x$, so linearity and shift invariance give
$$
\phi(x)=\phi(-x)=-\phi(x).
$$
Thus every such functional assigns the common value
$$
\boxed{\phi(x)=0},
$$
although the sequence does not converge.

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