A question from Asymmetric random walk

Question

I am studying the asymmetric random walk problem from the following notes.

https://nicfreeman1209.github.io/Website/MASx52/html/Hitting-probabilities-random-walks.html#lem:E_T_biased

I am bit struggling while understanding the proof of $E(T)<\infty$.

In the notes, a random variable $K$ is defined as $$ K = \inf\{k \in \mathbb{N}; E_k~~ occurs\} $$ is a geometric random variable with success parameter $p^m$ and $P(E_k)=p^m$. This means $K < \infty$ a.s. and that $E(K)=p^{-m}< \infty$.

I am confused why $E(K)=p^{-m}$? Can anyone please exaplain this step? And

If $E(T) < \infty$, then how to show $P(T<\infty)=1$?