How is arthur convinced that Lucy is un-dead and must be killed?

Arthur is convinced that Lucy is un-dead and must be killed after witnessing her increasingly vampiric behavior and the mysterious deaths occurring around her. He is further persuaded by Dr. Seward and Van Helsing, who provide evidence of her transformation into a vampire, including the bite marks on her neck and her unnatural allure. Their shared observations about her nocturnal activities and the bloodlust she exhibits culminate in Arthur's realization that she poses a threat to others, prompting his decision to end her life to save her soul.