I need to determine if $((0, 1), d)$ where $d(x, y) = |x^2 - y^2| \forall_{x,y}\in (0,1)$
My argument is as follows, take the sequence defined by $\dfrac{1}{n}$, then we know by $d(x, y)$ that $\dfrac{1}{n} \to 0$ as $n \to \infty$. As $0 \notin (0, 1)$ and the sequence $\dfrac{1}{n}$ is Cauchy w.r.t $d(x, y)$ we have $((0, 1), d) is not a complete metric space by definition.
Also, is $\left(\left(\dfrac{-\pi}{2}, \dfrac{\pi}{2}\right), d \right)$ where $d(x, y) = |tan(x) - tan(y)|$ for all $x, y \in \left(\dfrac{-\pi}{2}, \dfrac{\pi}{2}\right)$
Can I just consider $\dfrac{\pi}{2} - \dfrac{1}{n}$? Then I know this sequence converges to $\dfrac{\pi}{2}$ which is not in $\left(\dfrac{-\pi}{2}, \dfrac{\pi}{2}\right)$.
- Could somebody highlight how the metric $d(x, y)$ influences the problem?