How can we define a Metric $d$ ( in space $\mathbb{R}$ of reals) so that sequence $A_n = 1/n, n = 1,2,3,...$ converges to a non-zero real number?

Question

How can we define a Metric $d$,in space $\mathbb{R}$ of real numbers,so that sequence $A_n = \frac 1 n$ , $n = 1,2,3,...$, converges to a non-zero real number ?

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score 2 · Accepted Answer · answered Nov 28 '20 at 14:55

Define $f : \Bbb{R} \to \Bbb{R}$ by: $$ \begin{align*} f(x) = \left\{ \begin{array}{l@{\quad}l} 1 - x & \mbox{if $x \in \{0, 1\}$}\\ x & \mbox{otherwise} \end{array} \right. \end{align*} $$ Then define a metric $d$ by: $$ d(x, y) = |f(x) - f(y)| $$ Under $d$, $\frac{1}{n} \to 1$ as $n \to \infty$.

How can we define a Metric $d$ ( in space $\mathbb{R}$ of reals) so that sequence $A_n = 1/n, n = 1,2,3,...$ converges to a non-zero real number?

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