The influencing style that the BA is using is inspiring. Inspiring is a style that appeals to the emotions and values of the stakeholders, and motivates them to achieve a shared vision or goal. The BA is using an inspiring style by showing them a video case study of a contract renewals team at another company that successfully adopted Sales Cloud, which shortened its time to close and increased its close rate by 30%. This helps to create a positive and aspirational image of what Sales Cloud can do for them, and encourages them to overcome their resistance and embrace the change. References:
https://trailhead.salesforce.com/en/content/learn/modules/business-analysis-stakeholder-management/influence-s