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Re: Online extension courses & PhD admissions
That's right - if it's the same professor I had, he likes to make proofs very central to the course. And exams (if I remember correctly) are largely based on memorizing a number of proofs. Like I said, it is not an easy course: it covers a lot of material at a pretty high level. You can judge best for yourself whether you would be able to perform well int hat environment.
You can find a bunch of the course materials and lecture videos online. For example, the main assistant for the course (initials KP) has many review section videos posted on YouTube that are easy to find. I recommend checking these resources over for a better sense of what to expect.
Regarding the online format, I would say resources are not a big concern. Harvard Extension does a solid job of recruiting a handful of TA's for the course who hold office hours and review sessions regularly via online video chats. They also make it easy to connect online with other students in the class. If you have internet problems, sure that would be a concern. But I would say otherwise the online factor likely does not detract much if at all from the course, and it is pretty easy to be in touch with TA's or other students.