Jiayin Lin

Freshman intended math+cs major.
Welcome to discuss 104 problems or other math/cs with me ^^ happy to learn from everyone!!
Courses I have taken:
math W128A
math 53
math 54
cs 61A
Current Course:
math 104
math 113
math 110
cs 70

Peano axiom for $\mathbb{N}$ produce axiomic instead of constructive definition, but how can we know that the $\mathbb{N}$ that satisfies all the axioms is unique?
Is it that we use these axioms to define a group of sets that can be considered “natural numbers”, and then use induction to show that A and B that satisfies all axioms must be equal? (since we already have inductive property for both A and B now)
I finally understood how to upload Ross 3.2 to this site now..

Real number has two defs, but textbook did not show the equivalence and their equivalence to completeness… HW is here

HW for sequence and limits I did not finish 10.11 I found an inf for the sequence to be 1/2 but I can no longer proceed further…

Got a hint from a textbook to evaluate $\int_0^{\frac{\pi}{2}}\sin^{2n+1}x dx$ first, now it worked but literally dont understand how we are supposed to think about this at the first place :(

Subsequential limits are difficult for me.

During class there are 2 that I did not follow at first:

1: there is a subsequence approaching $limsup(s_n)$ monotonically.

2: the set of subsequential limit of a bounded sequence is closed (This one is better explained just by a graph)

HW3 is quite easy this week though, but cantor diagonal is a really amazing technique.

Series is no good for me.

I spent my whole life on the last question in rudin in this week HW4.

Here are my 5 questions:

In our proof, divergence in quotient and root test both implies an almost-geometric sequence in the tail. I really dont understand how is it helpful in determining most of the non-obvious diverging series, since we usually dont need to spend much effort to show an increasing sequence diverge. :(

In hw I see for a positive sequence, the fact that its partial sum is a monotone sequence can be helpful to see its convergence. Is there a lot of instances that we can try to “zip” a sequence into positive parts, and then try to bound the partial sum? (its really just an intuition that may not be useful at all)

When we try to show some convergence and divergence of some series expressed as fractions (i.e. $\frac{n+1}{n^3+1}$) can we just compare the power terms and say it is convergent? (since 1/n^2 is) or this only works for sequence.

When we want to analyze the set of subsequential limits, it there anything else than limsup and liminf that we can do…?

if we can show that a sequence is bounded by the max of two sequences that has divergent series, can we say the series diverges? (something like comparison test)

MT went smooth although I made stupid mistakes.

Metric space is really hard, even at the beginning. Here is my HW 5, I spent a lot of time on Ross 13.17.

It is really smart that someone reminded me of taking a Q inside each interval, since they are disjoint they are all unique, and thus there is a bijection.

I have done HW 6 but the last question I do not get it.

I see that cantor set can be a valid counter example, but I still dont understand how the given statement is wrong. It seems a really valid bijection from every open interval to its left (infimum).

hw7 was easy, but the first question I think there must be better ways for it. Mine was literally all over the place…

The question in class in interesting that every path-connected set is a connected set. I did not have time to get to that one during discussion, but my idea was that if the set $A$ is not connected, then $A=B \cup C$, B and C are clopen, nonempty and disjoint. Then take an element $x\in B$ and $y\in C$, if $f: [0,1]\to A$ satisfies $f(0)=x$ f is continuous, then $f^{-1}(B)$ is clopen and contains 0. However, we know [0,1] is connected, so the only nonempty clopen subset is itself, so $f([0,1])=B$, which means $f(1)\not=y$.

hw 8 is easy for me, but I dont quite get the Weierstrass-M test since for me I feel it is a too strong requirement that sup of all $|f_n|$ is bounded by something that sum up to converge then the the convergence is too obvious. I am concerned about what happens when $\sum\limits_m^n sup(|f(x)|)$ is not cauchy (you are adding up values for different x in each term) but $sup(\sum\limits_m^n |f(x)|)$ converge to 0 for all x and the series is still uniform cauchy and how we identify this sort.

I did horribly in MT2, so I have to study harder. HW9 is here.

I did the first question wrong, but I figured out that integrate my answer will work. This one was really cool.

This was my original answer I thought I was constructing some that vanish at both points

This works now properly

I had a really hard time going through R-S integral. I missed a class and I cannot follow anymore… :(

I have been frantically going through Ross 35 but I found there are so many material.

I did hw10 with no ease, and I hope they are correct.

This hw11 is quite easy for me though the last one I took some time to come up with, because I am too far from being familiar with R-S integral.