### Metric Spaces

##### due by Thursday, Sep 8, 2022

Let $(X, d)$ be a metric space. Define a function $\bar{d}: X \times X \to \R$ by $\bar{d}(p, q) = \min(d(p, q), 1)$. Recall that the metric space $(X, d)$ is said to be bounded if there exits $M\in \R$ such that for all points $p, q\in X$ we have $d(p, q) < M$

1. Show that $\bar{d}$ is a metric on $X$.
2. Prove or disprove: The metrics $d$ and $\bar{d}$ induce the same topology on $X$.
3. Prove or disprove: The metrics $d$ and $\bar{d}$ are Lipschitz equivalent if and only if $(X, d)$ is bounded.