Do red dwarfs eventually become white dwarfs?

According to this image: http://www.siprep.org/faculty/aokeefe/images/stellarevolution-stellar_fate_type1a_label_300dpi.jpg ..a red dwarf will eventually degenerate into a white dwarf once it no longer has hydrogen left to fuse, but when I read about white dwarfs from various sources it always seems to imply... show more According to this image:
http://www.siprep.org/faculty/aokeefe/im...

..a red dwarf will eventually degenerate into a white dwarf once it no longer has hydrogen left to fuse, but when I read about white dwarfs from various sources it always seems to imply that white dwarfs come only from red giants. Can anyone please clarify this for me?
Update: Yes thresher I've read that, it doesn't seem to answer my question.
Update 2: No ron I didn't mean red giants, since I know very well that white dwarfs form from them. I said that in my question. Did you look at the image?
Update 3: Phoenix and ronwiz: thank you very much for your answers, but they appear to contradict each other so I'm know nearer knowing the answer :(
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