Evaluating min/max probability with Geometric random variables

Suppose that ${X}_{1},\dots ,{X}_{n}$ are independent, ${\text{geometric}}_{1}(p)$ random variables. Evaluate $P(\text{min}({X}_{1},\dots ,{X}_{n})>l)$ and $P(\text{max}({X}_{1},\dots ,{X}_{n})\le l)$

The textbook solution is as follows:

$\begin{array}{rl}P(\text{min}({X}_{1},\dots ,{X}_{n})>l)& =P(\bigcap _{k=1}^{n}\{{X}_{k}>l\})\\ \text{}& =\prod _{k=1}^{n}P({X}_{k}l)\\ \text{}& =\prod _{k=1}^{n}{p}^{l}\\ \text{}& ={p}^{ln}\end{array}$

and $\begin{array}{rl}P(\text{max}({X}_{1},\dots ,{X}_{n})\le l)& =P(\bigcap _{k=1}^{n}\{{X}_{k}\le l\})\\ \text{}& =\prod _{k=1}^{n}P({X}_{k}\le l)\\ \text{}& =\prod _{k=1}^{n}(1-p{)}^{l}\\ \text{}& =(1-{p}^{l}{)}^{n}\end{array}$

I'm having trouble understanding pretty much the entire solution. If I were to summarize what's particularly puzzling me:

1) How was the first line of each solution derived (i.e. $P(\text{min}({X}_{1},\dots ,{X}_{n}>l)=P({\cap}_{k=1}^{n}\{{X}_{k}>l\})$ and the corresponding equation for max)?

2) The PMF for ${\text{Geometric}}_{1}(p)$ is given as ${p}_{X}(k)=(1-p){p}^{k-1}$. How is $P({X}_{k}>l)={p}^{l}$? The same applies for the max case as well.

Suppose that ${X}_{1},\dots ,{X}_{n}$ are independent, ${\text{geometric}}_{1}(p)$ random variables. Evaluate $P(\text{min}({X}_{1},\dots ,{X}_{n})>l)$ and $P(\text{max}({X}_{1},\dots ,{X}_{n})\le l)$

The textbook solution is as follows:

$\begin{array}{rl}P(\text{min}({X}_{1},\dots ,{X}_{n})>l)& =P(\bigcap _{k=1}^{n}\{{X}_{k}>l\})\\ \text{}& =\prod _{k=1}^{n}P({X}_{k}l)\\ \text{}& =\prod _{k=1}^{n}{p}^{l}\\ \text{}& ={p}^{ln}\end{array}$

and $\begin{array}{rl}P(\text{max}({X}_{1},\dots ,{X}_{n})\le l)& =P(\bigcap _{k=1}^{n}\{{X}_{k}\le l\})\\ \text{}& =\prod _{k=1}^{n}P({X}_{k}\le l)\\ \text{}& =\prod _{k=1}^{n}(1-p{)}^{l}\\ \text{}& =(1-{p}^{l}{)}^{n}\end{array}$

I'm having trouble understanding pretty much the entire solution. If I were to summarize what's particularly puzzling me:

1) How was the first line of each solution derived (i.e. $P(\text{min}({X}_{1},\dots ,{X}_{n}>l)=P({\cap}_{k=1}^{n}\{{X}_{k}>l\})$ and the corresponding equation for max)?

2) The PMF for ${\text{Geometric}}_{1}(p)$ is given as ${p}_{X}(k)=(1-p){p}^{k-1}$. How is $P({X}_{k}>l)={p}^{l}$? The same applies for the max case as well.