The most straight-forward way of showing something is invertible is simply find an inverse element and check that it is indeed the inverse. I think there is a very good candidate for the inverse in this case.
As a rising junior in high school, I think you're in a good shape in terms of determining your major. I think it's important to pick a "direction" before you enter college (e.g. not specific enough to claim a major, but something along the line of "science," "humanity," "social science," or...
A good point. One more thing to add to my previous comment---Calculus I is a very sequential course, meaning, the material you learn in the course will build up what you have learned previously. It's like building an apartment complex with multiple stories---you first build the ground floor...
I think the hardest part of the book is actually the second chapter, which is on metric spaces. I would do my best to avoid metric spaces and basic topology from that book for the first time, especially about the compact sets. As soon as the definition of a compact set is given (in terms of open...
First, at least give your best shot for the final exam---some instructors like to bump up a grade of someone who showed some improvement on the final ("if you can do well on a cumulative exam, perhaps the student should deserve more than a failing grade"). Of course, not every instructor has...
Also, make sure what class those reviews are coming from. I know one good prof with 20 low ratings simply because there was one semester he had to teach a business calculus course with 500 students in it.
If you are not sure, then you should probably take a little bit of both, and that "little bit" depends on how flexible your schedule/program is (e.g. some math departments can let you take whatever you want and still graduate, whereas the others want to take you specific number of courses from...
Thanks! Based on the course descriptions, Analysis 2 sounds like a very reasonable sequence that follows Analysis 1. It can hard in a sense that the materials build upon what you learned in analysis 1 (afterall, you need to be solid on the analysis of R^1 in order to learn the analysis of R^n)...
I'd say Kosniowski is more elementary than Massey or Munkres. It is brief, and cover less material than the other two, but it does have a pretty good exposition of the subject, with a good balance of abstract idea and concrete examples.
Massey should be good if you already familiear with some...
Getting any PhD from a decent math program, including MIT, is challenging. I don't think this is something that we can easily compare between two schools (as definition of "harder" differs from a person to another person).
Getting INTO a PhD program in mathematics at MIT, on the other hand, is...
I think so too :wink: I now realize that I just happened to be a bad neighborhood where my point happened to be a maximum, and it took some work to walk away from that neighborhood.
(Of course, local max can be a global max, but most people probably won't reach there anyway...)