This is the 1549th most frequent Japanese word.
住み
"住み" means "living" or "residing."
Here, "住み" (as part of "住んでいます") is used to express ongoing habitation in a location.
私は東京に住んでいます。
I live in Tokyo.
In this sentence, "住み" (as part of "住み始めました") conveys the commencement of living in a place.
彼は新しいアパートに住み始めました。
He started living in a new apartment.
This usage of "住み" (as part of "住み続ける") demonstrates the continuation of residency in a particular place.