How do you know for sure whether or not to use a closing? One good test is to ask yourself whether this email is more akin to an instant message/text or a business letter. If you're giving your teammate a quick update on an ongoing project, a formal closing might not be necessary; if you're throwing your hat in the ring for a promotion, it definitely is required. When all else fails, and you're still not sure, err on the side of caution and include it. You'll never go wrong by being too polite and respectful.
Then you arouse their desire by explaining how you can do this — for instance, by showing how successful you’ve been in other positions, or by explaining the specialized training you have relevant to the job. Avoid jargon here — explain clearly and concisely the benefits of the solution you offer. When I worked in marketing, I learned an important lesson: nobody buys anything based on features (and “buy” is used in its loosest possible sense, like you “buy” an idea). You have to explain exactly how those features will benefit the potential buyer.