Let $d(x,y)=|x-y|$ and $\tilde{d}(x,y)=\frac{|x-y|}{\sqrt{(x^2+1)(y^2+1)}}$ be two metrics on $\mathbb{R}$. It is easy to see that $(\mathbb R,d)$ is complete but $(\mathbb R, \tilde{d})$ is not. Moreover $\tilde{d}(x,y) \leq d(x,y), \forall x,y\in \mathbb R$ and therefor $x_n\to x$ with $d$ then $x_n\to x$ with $\tilde{d}$. My question is, is there any sequence that converges with $\tilde{d}$ but does not converge with $d$?
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