Real Analysis 101: Tips for Writing Good Proofs

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Real analysis requires a different approach to problem-solving compared to engineering mathematics, focusing on abstract concepts and rigorous proofs. To improve proof-writing skills, it's recommended to study examples and work through proofs alongside practical applications. The book "Introduction to Real Analysis" by Bartle may feel too formal for beginners, prompting a request for more accessible resources. Participants suggest looking for beginner-friendly texts that simplify complex ideas. Engaging with examples and alternative literature can significantly enhance understanding and proficiency in writing proofs.
Rakiztah
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hello everyone!
I just started a course in real analysis and i must say that it is quite different from all the "engineering math" that i have taken before.I was wondering if anyone could give me tips or advice on how to get better at writing good proofs. Right now,we are using a book called Introduction to real analysis by Bartle,but its too formal for my taste. Also,if anyone could suggest a book that is made for a total beginner,that would be much appreciated.
Thanks in advance.
 
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Proofs are by nature more abstract and different than engineering problems. Best you can do is choose some examples and follow the proof along such examples in parallel. A standard book about analysis is
https://www.amazon.com/dp/0914098918/?tag=pfamazon01-20
(older editions are cheaper)
 

Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
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