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Oct 8, 2020 at 7:26 history edited Algernon CC BY-SA 4.0
Adding the update
Oct 7, 2020 at 21:12 comment added Algernon You are right, I missed the fact that $S$ is supposed to be the span of $p_1,p_2,\ldots$.
Oct 7, 2020 at 20:59 comment added Pietro Majer Note that $S$ is the span of the $p_k$, so there are many other converging sequences $g_n$ in $S$ (for instance $S$ could be the whole ring $\mathbb{R}[x]$, in which case the OP statement is true), so it seems more complicated (I also think it's not true, for similar reasons)
Oct 7, 2020 at 20:45 history answered Algernon CC BY-SA 4.0