Elementary Real Analysis L1 2020.09.07

Elementary Real Analysis L1

The difference between the calculus and real analysis is one only requires using and know how to solve problems by calculus, while the other need you know why one can use it.

There are some example for wrongly using theorems:

1. Ex:
   S=1-1+1-1+1… Calculate the sum of S
   ~S=(1-1)+(1-1)+(1…=0
   ~S=1+(-1+1)+(-1+1)+(-1+1)… =1

2. Ex:
   $\lim_{x\to0} \lim_{y\to0} \left( \frac{x^2}{x^2+y^2} \right)=1 \lim_{y\to0} \lim_{x\to0} \left( \frac{x^2}{x^2+y^2} \right)=0$x→0lim​y→0lim​(x2+y2x2​)=1y→0lim​x→0lim​(x2+y2x2​)=0

3. Ex:
   当多边形的直角越来越多，多边形周长越来越近似于2pi*r，但多边形的周长始终保持为8r