Ceiling of a metric is a metric Suppose $d$ is a metric on the set $X$. For $x$, $y \in X$ define the function $c$ by $c(x,y) = \lceil d(x,y) \rceil$. Show that $c$ is a metric on $X$.
(The reason I put the question here is because I was wondering if it was possible to "discretise" a metric, and the simplest way that I thought of was to use the ceiling.)
 A: To be a metric, $c$ must satisfy the following three conditions:


*

*$c(x,y) \geq 0$ for all $x$, $y \in X$.

*$c(x,y) = 0$ if and only if $x=y$.

*$c(x,y) = c(y,x)$ for all $x$, $y\in X$.

*$c(x,z) \leq c(x,y) + c(y,z)$ for all $x$, $y$, $z \in X$ (the triangle inequality)


Before we start considering these four conditions, let us consider the properties of the ceiling function $t \mapsto \lceil t \rceil$ for $t \geq 0$. 
The ceiling function is non-decreasing, so if $s \leq t$ then $\lceil s \rceil \leq \lceil t\rceil$. 
Also, given two numbers $s$ and $t$ with $s, t \geq 0$, $\lceil t \rceil$ is the smallest integer greater than or equal to $t$, so we have that $t \leq \lceil t \rceil$ and similarly $s \leq \lceil s \rceil$. Thus $s + t \leq \lceil s \rceil + \lceil t \rceil$. However, $\lceil s + t \rceil$ is the smallest integer greater than or equal to $s+t$ and $\lceil s \rceil + \lceil t \rceil$ is an integer greater than or equal to $s+t$. Therefore it must be that $\lceil s+t \rceil \leq \lceil s\rceil + \lceil t\rceil$.
Now we consider the four conditions in turn:


*

*Since $d(x,y) \geq 0$, $\lceil d(x,y) \rceil  \geq \lceil 0 \rceil = 0$. So $c(x,y) \geq 0$.

*$c(x,y) = 0$ if and only if $\lceil d(x,y) \rceil = 0$, if and only if $d(x,y) = 0$ (since $d(x,y)$ is nonnegative), if and only if $x=y$.

*$c(x,y) = \lceil d(x,y) \rceil = \lceil d(y,x) \rceil = c(y,x)$ for all $x, y \in X$.

*Given $x, y, z \in X$, 
$$
\begin{align}
d(x,z) &\leq d(x,y) + d(y,z)\\
\lceil d(x,z) \rceil &\leq \lceil d(x,y) + d(y,z) \rceil\\
&\leq \lceil d(x,y) \rceil + \lceil d(y,z) \rceil
\end{align}
$$
So $c(x,z) \leq c(x,y) + c(y,z)$.
Therefore $c$ is a metric on $X$.
