Before my own depression was diagnosed, I dated a man who suffered from clinical depression and alcoholism. Of course, I wasn't aware of this when I started dating him, or I never would have started. I don't have a burning need to "fix" people. A couple of months after I started treatment for my depression, we split up. Although I think it was more or less mutual, I would not have stayed around for long in any case. I had gotten tired of trying to talk him into getting help for his depression. He had kept the alcoholism at bay by quitting drinking cold turkey, but the underlying problem, the depression, was still there. For some reason, he was dead set against any kind of treatment for his depression. He had had therapy when he was in his teens, and apparently it had done more harm than good.
I knew very little about clinical depression at that point, or I might have been able to come up with a more effective strategy to persuade him to get treatment. Although I had had depression for almost two decades, until I was diagnosed, I didn't recognize certain behaviors of mine for what they were. So here's my "if I knew then what I knew now" insight into what might work to persuade someone to get treatment for depression.
With many, if not most, people with depression, there are usually a few forces at work. I think the two most effective in terms of keeping someone from getting help are fear that no one can help and apathy - a lack of both mental and physical energy.
I'd say the most pervasive, strongest symptom of depression that gets in the way of getting treatment is hopelessness. A complete lack of hope is one of the hallmarks of depression. Until you've experienced this, it's hard to understand how important hope actually is to us. It's what keeps us going when we don't have any concrete evidence that we're going to succeed. Without hope, everything seems unsurmountable. That's where the fear that no one can help comes in. Even someone who is open to the idea of depression treatment is sometimes terrified, more than anything, that the doctor will say that there's nothing that can be done. Their lack of hope won't let them believe that there is a good chance that they can be helped.
In this case, what you can say to them is, "I know you may think that nothing can be done, but did you know that over 80% of people with depression can be successfully treated?" This is where it's important to do your research, print out information about depression treatment and, if appropriate, go over it with the individual.
The physical and mental exhaustion that comes from depression makes it difficult enough to do something simple like return books to the library on time, let alone take the steps necessary to find a doctor and make an appointment. With the depressed person's permission, do the research into finding a couple of good doctors, including whether they take the right insurance. Then make some time to help the individual actually make the appointments. If they want you to come along for moral support and you feel comfortable doing it, that's definitely a good idea. Remember, depression can make your mind pretty cloudy, and they may appreciate your recall of the appointment afterwards, which will probably be better than theirs.