Can I prove my database does not have memory issues?
The only way to prove is not a memory issue is by prooving is something else. And this requires you to identify the root cause of your performance problems. I recommend you follow a methodology like Waits and Queues. The SQLCAT team has also published a Troubleshooting Flowchart poster that you can follow.

As a general side comment: if someone is offering you to buy more memory on the server, just say YES, and then go ahead and do the root cause analysis. It doesn't matter that if the root cause of the problems are table scans, SQL Server can always use more memory.