Ending a sentance with iru or aru after the verb?

Ending a sentance with iru or aru after the verb?
Hello! Thank you for this forum, it is very informative and the discussions are intriguing! I do try to search out my questions before I post one of my own, but this one is tricky because I keep getting search results about the difference between Godan and Ichiban verb... haha. So on to my question!

I am learning the grammar from Tae Kim as I work my way through "Japanese Core 2000" anki deck,and I noticed something strange... at the end sometimes the verb is finished with a form of いる:

1) 彼は会社の近くに住んでいる。

2) 電車で子供が騒いでいた。

3) 宿題は未だ終わっていません。

and last my favorite:

4) 皆一生懸命生きている。

The fourth sentence makes me chuckle because I feel like 一生懸命 is a word with large pants while 生きて essentially means a similar thing but less detail and than いる and you have the simplest form... do you see what I am saying? I know they are all a little different but the sentence seemed to work its way down from complexity.

My point:
Why end with a form of iru?

Thanks,
Craiggles.

The ている form is also called the present progressive form. It basically is a bit like adding -ing in English.

Examples are

食べる - to eat. 私はりんごを食べる I eat apples
食べている - eating. 私はりんごを食べている I am eating an apple.

So your favourite number 4 is something like.... Everyone is living as best they can or with all their might etc.

Enduring state-of-being vs enduring state of action
Other uses of the te-form | Learn Japanese

As for #3, 終わっています describes that the current state is resulted from the action "to finish your homework", i.e., you have finished your homework already. 未だ終わっていません is the negative of it, thus, you haven't finished your homework yet.