One of the very nice features of null scalar sequences is the fact that they admit summable subsequences. Is the same true in Banach spaces? That is, if $(x_n)_{n=1}^\infty$ is a sequence in a Banach space $X$ and $\|x_n\|\to 0$, must $(x_n)_{n=1}^\infty$ have a summable subsequence?
2026-04-21 13:25:50.1776777950
Do null sequences in Banach space have summable subsequences?
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Yes, take a subsequence $(x_{n_k})_k$ such that $\|x_{n_k}\|\leq 2^{-k}$. Then $$\|\sum_{k=1}^{m_2}x_{n_k}-\sum_{k=1}^{m_1}x_{n_k}\|=\|\sum_{k=m_1+1}^{m_2}x_{n_k}\|\leq\sum_{k=m_1+1}^{m_2}\|x_{n_k}\|\leq\sum_{k=m_1+1}^{m_2}2^{-k}=2^{-m_1}-2^{-m_2}\to0$$ as $m_1,m_2\to0$. Thus the partial sums of $x_{n_k}$ are Cauchy.