Let $(X,d)$ be an unbounded metric space. Is it right to say: There are a $c\in X$ and $\{x_n\}_{n \in {N}}\subset X$ such that $\lim_{n \to +\infty}d(x_n,c)=+\infty$?
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1Take $c\neq x_1$ to be any two points of the space. If all other points were inside the ball of radius $r_n=\max(n, d(c,x_n)+1)$. Then the distance between any of them would be not larger than $2r$, by triangle inequality. Therefore there is some point $x_{n+1}$ outside that ball. – logarithm May 26 '19 at 12:44
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