In the proof of the open mapping theorem in "Functional Analysis" by Rudin, there is the following argument:
Let $X$ be a topological vector space in which its topology is induced by a complete invariant metric $d$. Define $V_n = \{ x \in X : d(x,0) < 2^{-n}r \}$ for $n=1,2,\cdots$, where $r>0$. Suppose $x_n \in V_n$. Since $x_1 + \cdots + x_n$ form a Cauchy sequence, it converges to some $x \in X$ with $d(x,0) < r$.
In the above, I cannot understand why $d(x,0)
Any help would be appreciated.