You analyze a network of $$ mobile devices, with $$ even. The devices move around with their

owners. At any point in time, the devices form a graph, with each node representing each of

the $$ devices, and with an edge between device $$ and device $$ if the physical locations of $$

and $$ are no more than $1000$ meters apart $($ and $$ are $$in range$$ of each other.$)$ At all times,

each device $$ is within $1000$ meters of at least $/2$ of the other devices. Does this property

guarantee that the graph representing the network will remain connected at all times? Prove

your claim.