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What people mean by "real analysis" is a combination of the latter two courses. It seems to me that any student taking the second course (real analysis) should know the material of the first (principles of mathematical analysis). In my school, these two courses form a year long sequence (Real Analysis I and II).
Take "Principles of mathematical analysis". It covers most of the fundamentals.
P.S. The " Introduction to mathematical analysis" is called "Intro to higher math" in my school. It's a prerequisite for the real analysis courses, but you don't really need it.
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"Since it befalls, that in most instances
Current opinion leans to false: and then
Affection bends the judgment to her ply."
Dante Alighieri
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