In the book Functional Analysis of Rudin, theorem 1.15 (c) says that suppose $V$ is a neighborhood of 0 in a topological vector space $X$, if $\delta_1\geq\delta_2\geq\cdots$ and $\delta_n\to 0$ as $n\to\infty$, and if $V$ is bounded, then the collection $\{\delta_nV:n=1,2,3\cdots\}$ is a local base for $X$.
Why we can't make a conclusion from this that every topological vector space is first countable since we have found a countable local base.
Is the problem arise from that the countable local base is only of $0$?
