# Need a lot of worked real analysis proofs (from easy to difficult)

## Main Question or Discussion Point

I was accepted into a top tier Ph.D. Operations Research program. I have six months to prepare independently on my own (at home). Everybody told me real analysis is the first thing I should look at (which makes sense, because I don't have proof experience).

Can you please recommend me a book that has a high number of worked examples (e.g., any solutions manual).

I got the Bartle and the Rosenlicht text, but there are no solutions to the exercises, and in the text there are not many worked proofs.

Remark: I posed this thread in two forums because I want to have higher exposure and I didn't know which one would be more fitting.

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Since you posted this in 2 separate threads, the other one has been deleted, it was deleted exactly when I was clicking the "Post Quick reply" button, Lol!

Regarding the books, I would recommend "Berkeley Problems in Mathematics" by Paulo Ney de Souza & Jorge-Nuno Silva. It contains problems and solutions that covers all major topics in Math at undergraduate level (you may ignore Complex Analysis & Groups if you want to since they won't be of much use in Operations Research). The contents are:

1 Real Analysis
1.1 Elementary Calculus
1.2 Limits and Continuity
1.3 Sequences, Series. and Products
1.4 Differential Calculus
1.5 Integral Calculus
1.6 Sequences of Functions
1.7 Fourier Series
1.8 Convex Functions

2 Multivariable Calculus
2.1 Limits and Continuity
2.2 Differential Calculus
2.3 Intcgral Calculus

3 Differential Equations
3.1 First Order Equations
3.2 Second Order Equations
3.3 Higher Order Equations
3.4 Systems of Differential Equations

4 Metric Spaces
4.1 Topology of R^n
4.2 General Theory
4.3 Fixed Point Theorem

5 Complex Analysis
5.1 Complex Numbers
5.2 Series and Sequences of Functions
5.3 Conformal Mappings
5.4 Integral Representation of Analytic Functions
5.5 Functions on the Unit Disc
5.6 Growth Conditions
5.7 Analytic and Meromorphic Functions
5.8 Cauchy’s Theorem
5.9 Zeros and Singularities
5.10 Harmonic Functions
5.11 Residue Theory
5.12 Integrals Along the Real Axis

6 Algebra
6.1 Examples of Groups and General Theory
6.2 Homomorphisms and Subgroups
6.3 Cyclic Groups
6.4 Normality, Quotients, and Homomorphisms
6.5 S,. A,. D,
6.6 Direct Products
6.7 Free Groups. Products. Gcnerators. and Relations
6.8 Finite Groups
6.9 Rings and Their Homomorphisms
6.10 Ideals
6.11 Polynomials
6.12 Fields and Their Extensions
6.13 Elementary Number Theory

7 Linear Algebra
7.1 Vector Spaces
7.2 Rank and Determinants
7.3 Systems of Equations
7.4 Linear Transformations
7.5 Eigenvalues and Eigenvectors
7.6 Canonical Forms
7.7 Similarity
7.8 Bilinear. Quadratic Forms. and Inner Product Spaces
7.9 General Theory of Matrices

Do you mind to share your profile? I am pretty much interested in applying for Operations Research PhD myself!

Thanks a lot, this is much more than I expected: I need to also cover rigorous linear algebra.

Not sure what you mean by profile. If you mean qualifications by that: undergrad GPA: 3.97, math GRE (not the subject test): 800.

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gb7nash
Homework Helper
Baby Rudin is pretty good.

Thanks for all the recommendations.

If its for Real Analysis then I would recommend Real Mathematical Analysis by Pugh. It doesn't have solutions, but the text is very well written. If it is for an introductory course in Analysis I would recommend Calculus by Spivak followed by Spivak's Calculus on Manifolds or Analysis on Manifolds by Munkres.

lavinia
Gold Member
I was accepted into a top tier Ph.D. Operations Research program. I have six months to prepare independently on my own (at home). Everybody told me real analysis is the first thing I should look at (which makes sense, because I don't have proof experience).

Can you please recommend me a book that has a high number of worked examples (e.g., any solutions manual).

I got the Bartle and the Rosenlicht text, but there are no solutions to the exercises, and in the text there are not many worked proofs.

Remark: I posed this thread in two forums because I want to have higher exposure and I didn't know which one would be more fitting.
A lot of OR involves probability theory and statistics. A good book on stochastic processes and read papers on applying stochastic models to real world problems e.g using kalman filters. There is a huge literature in finance.