How can i become good in proving theorems in real analysis?
Answers
Answer:
it's just plain harder, the way you learn real analysis is not by memorizing ... in your own proofs. The best way to get good at this is to take your time; read slowly,.
Step-by-step explanation:
please mark it as brainlist
Most of the theorems in real-analysis (especially those in introductory chapters) are intuitive and based on the concept of inequalities. If one understands the concept of inequalities (not in the sense of memorizing AM greater than equal to GM or other famous inequalities) in terms of comparison of numbers most of the common proofs are trivial applications of the definitions.
I will provide two examples:
1) If ff is continuous at x=ax=a and f(a)f(a) is positive then there is a neighborhood of aa in which ff is positive.
Now one has to know what is meant by continuity to prove this. Informally this means that values of f(x)f(x) are arbitrarily near f(a)f(a) if xx is sufficiently near aa. The ϵ,δϵ,δ are used to quantify "arbitrarily" and "sufficiently" in a formal manner. Now if we see that f(a)f(a) is positive then there is a range of values near f(a)f(a) which are positive. Hence if xx is sufficiently close to aa, f(x)f(x) will take values in the range near f(a)f(a) and these are all positive as mentioned in last sentence.
2) If f(x)≤g(x)f(x)≤g(x) in a neighborhood of aa and both limx→af(x),limx→ag(x)limx→af(x),limx→ag(x) exist then