How did the Eastern Romans try to Retake their former Empire?

While the Western portion was on a rapid decline during the 4th and early 5th centuries, the East was doing alright. Thus, when the Western Roman Empire completely collapsed before the start of the 6th century, a burning question quickly arose: would the Byzantines attempt to reclaim the fallen half of the once-united empire? At first, no, but the succession of Justinian I to the throne in Byzantium would suddenly change that answer…