Given the following steps of a Map$-$Reduce job, order them in correct sequence that they are executed in: Partition, Combine, Map, Shuffle, Reduce. Which steps are guaranteed to execute? Which are not?$5\mathrm{.}40$ Prove Theorem $5\mathrm{.}5\mathrm{.}12$ for the case where $x_n$ and $x$ are continuous random variables.

$($a$)$ Given $t$ and $,$ show that $P(xt-)P(x_{n}t)+P(|x_n-x|).$ This

gives a lower bound on $P(x_{n}t).$

$($b$)$ Use a similar strategy to get an upper bound on $P(x_{n}t).$

$($c$)$ By pinching, deduce that $P(x_{n}t)P(xt).$