As MasterMind said, the same thoughts can be applied to reality. We only THINK we're lucid in reality. We're convinced of it, we have memories of thinking clearly in reality. The same thing is true for a lucid dreamer's lucid dreams. They are convinced they were thinking clearly. So what's the difference? As long as it FEELS like we can think clearly, does it matter at all if that's actually true or a false memory? No, it doesn't, just like we don't trouble ourselves with whether we are actually lucid in waking life.

Besides, it's quite easy to prove to yourself that you are lucid. I doubt the unconscious mind can do mental arithmetic very well - something that is possible when you're lucid, and a technique I use to strengthen my lucidity. The first couple of times I did it I made extra effort to record what calculations I did and the results of those calculations in my memory. Upon waking up, what do you know? I was correct.
Another good way of proving lucidity to yourself while you're dreaming is remembering your waking life. In my experience this has a tendency to destabilize the dream quite significantly, but it's possible. You remember everything up until you go to bed that night. This feels really conclusive as evidence that right now, you are dreaming, and it gives you a kind of continuity from your waking awareness. It's a really powerful feeling in my experience.