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Given the sequence in $\mathbb{R}^\omega$: $$y_1=(1,0,0,0\ldots), y_2=(\frac{1}{2},\frac{1}{2},0,0\ldots), y_3=(\frac{1}{3},\frac{1}{3},\frac{1}{3},0,\ldots),\ldots$$ I know that it converges in the product topology because it converges pointwise to $(0,0,\ldots).$ I know that it converges in the uniform topology since it converges to $(0,0,\ldots)$ uniformly. How do I determine whether or not this sequence converges in the box topology?

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up vote 6 down vote accepted

Let $$U=\prod_{n\ge 2}\left(-\frac1n,\frac1n\right)\;.$$ Clearly $U$ is an open nbhd of $\langle 0,0,0,\dots\rangle$; does it contain a tail of the sequence $\langle y_n:n\in\Bbb Z^+\rangle$?

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