True. But it's also well known that Alzheimer's increases the buildup of amyloid-beta, which latches into hippocampus (memory) and thalamus (which controls sleep). With lack of sleep, your brain is unable to get rid of the excess of amyloid-beta.
It's unknown if lack of sleep causes the onset of Alzheimer's, but the opposite is likely true: Alzheimer's patients don't sleep well. So once it starts to develop, it spirals down likely because of sleep deficiency, thus accelerating memory loss.
Source: "Why We Sleep", by Matthew Walker. We had a few threads recently about the book here [1], [2].