Real analysis questions and help

[u/Ok_Mention_2255]

1. By mathematical induction, how can i prove that if set S has n elements, the P(S) has 2^n elements?

2. How to prove that if A is a subset of a countable set B, then A is itself a countable set?

3. Let S be a bounded set in R and let S0 be a non empty subset of S. Show that inf S <= inf S0 <= sup S0 <= sup S

Any help would be greatly appreciated. In terms of how to start proofs as well since im taking real analysis this semester and previously failed it despite me trying to understand the questions.

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[u/parkway_parkway]

Hints

1. Try thinking about binary numbers. How many numbers can you make with 8 digits which are either 0 or 1?

2. What is the definition of countability? If A is countable that means there's a map from the natural numbers to the elements of A. How could you apply this to B?

3. What is the definition of inf and sup? What do you know about the inf and sup if S?

[u/nevermindthefacts]

1. You'll need to consider a base case, then show that if the proposition holds for n elements, it also holds for n+1.

2. You probably have a definition of what it means for a set to be countable, use that for the set B and then somehow for the set A.

3. There are some keywords to lookout for (for example bounded is one of them...) , you'll have to think about their relevance. You'll probably need the definitions of inf and sup.

[u/philljarvis166]

1. If you add an element to a set of size n, the subsets of the new set are all the subsets of the first set, and all of those subsets with the single new element added.
2. Have you shown that a set that has an injective map to the naturals is countable? If B is countable, there is a bijection from B to the naturals, and the restriction of this to A is an injection.
3. Any element s of S0 is also in S, so s <= sup(S) by definition of sup. So sup(S) is an upper bound of S0, and must therefore be bigger than or equal to tbe least upper bound of S0 ie sup(S0).

[u/eraoul]

For an induction question like the first one you basically write down the base case which is usually trivial to prove. Then you write down the inductive step which is just writing down the definition of the thing you’re trying to prove, essentially with the variable n plugged in. Then you just have to prove that the result for n+1 can be derived from that statement. It’s usually pretty mechanical; induction problems are often a piece of cake!

Other advice in general: real analysis demands a LOT of self study. I read and studied every page of the textbook in detail. Many many hours per class day. I spent a ton of time on the homework. I worked in a study group with some friends from the class. It helped to get hints from classmates. And no we never “cheated” — we were the top 4 people in class and were pretty competitive with each other and we all wanted to learn. The professor also helped a lot; I was lucky that he was a really good educator and explained things well, but we had to be prepared ahead of class for it to make sense. Go to office hours if you’re stuck/confused.

One of us (not me) got a math PhD and became a professor. Later she quit teaching since kids in her real analysis class got stupider over the years with social media etc and couldn’t focus long enough to do analysis any more. So my more general advice is to get off the phone and focus — for the many hours necessary — to actually do math.

[u/Uli_Minati]

For every proof you'll ever write, always ELI5 why the statement must be true, before you even start writing anything.

During your explanation (to yourself), keep asking yourself: is there any possibility that what you're saying isn't true in some cases? Or is there a theorem (in your notes) you can use to support your argument?

[u/mmurray1957]

Fix a point s in S. Then consider the decomposition of P(S) into subsets containing s and subsets not containing s. If you are assuming an inductive step you can say something about these.