Book Title: Understanding Analysis Author: Stephen Abbott Edition/Year: 2nd edition, 2015 Read: Chapters 1 ~ 7 (8 chapters in total) Exercises Solved: All (Chapters 1 ~ 7)
When I picked up Abbott's "Understanding Analysis", I wasn't sure if real analysis would be an interesting subject. Having only basic set theory under my belt, I thought it would be just "calculus done rigorously" - formalism for the sake of formalism. But little did I know, this would be the beginning of my first true mathematical love )).
"Understanding Analysis", despite being a rigorously written textbook, almost reads like a conversation between you and the author. Every chapter begins with a "Discussion" section that motivates new ideas. For example,when introducing epsilon–N proofs, Abbott doesn’t just drop definitions - he shows why they matter through examples of pathological sequences and series, like rearrangements that change the sum or limits that don’t commute.
I studied the book almost every day for six months, working carefully through Chapters 1–7 (I’ve left Chapter 8 for the future). Since this was my first real higher math text, I often struggled with the execution of proofs: summations, inequalities, or inventing a clever function at the right moment. The exercises on power series almost broke me - I remember spending 3 days trying to show that where 1/L is the radius of convergence. When the reasoning finally fell in place, I had a newfound respect for power series.
However, more than the theorems, what I took away was a sort of mathematical patience. I learned that being stuck in mathematics is a natural process of learning, and the important thing is not to "never get stuck", but to know how to "unstuck" yourself. That lesson, is worth just as much as any epsilon-delta proof. It also taught me how to think like an analyst: to search for hidden leverage in the problem’s assumptions and exploit them.
Moving forward, I'm now studying from John and Barbara Hubbard’s Vector Calculus, Linear Algebra, and Differential Forms. But I carry with me the conviction that I can learn real mathematics independently, as long as I'm willing to wrestle with the ideas. If you're hesitating on whether to pick up Abbott's Understanding Analysis, my advice would be to brush up on proof techniques, and dive in. It will be difficult, but finishing it might just make you fall in love with analysis.
Math Textbook Recommendation
Book Title: Understanding Analysis
Author: Stephen Abbott
Edition/Year: 2nd edition, 2015
Read: Chapters 1 ~ 7 (8 chapters in total)
Exercises Solved: All (Chapters 1 ~ 7)
When I picked up Abbott's "Understanding Analysis", I wasn't sure if real analysis would be an interesting subject. Having only basic set theory under my belt, I thought it would be just "calculus done rigorously" - formalism for the sake of formalism. But little did I know, this would be the beginning of my first true mathematical love
)).
"Understanding Analysis", despite being a rigorously written textbook, almost reads like a conversation between you and the author. Every chapter begins with a "Discussion" section that motivates new ideas. For example,when introducing epsilon–N proofs, Abbott doesn’t just drop definitions - he shows why they matter through examples of pathological sequences and series, like rearrangements that change the sum or limits that don’t commute.
I studied the book almost every day for six months, working carefully through Chapters 1–7 (I’ve left Chapter 8 for the future). Since this was my first real higher math text, I often struggled with the execution of proofs: summations, inequalities, or inventing a clever function at the right moment. The exercises on power series almost broke me - I remember spending 3 days trying to show that where 1/L is the radius of convergence. When the reasoning finally fell in place, I had a newfound respect for power series.
However, more than the theorems, what I took away was a sort of mathematical patience. I learned that being stuck in mathematics is a natural process of learning, and the important thing is not to "never get stuck", but to know how to "unstuck" yourself. That lesson, is worth just as much as any epsilon-delta proof. It also taught me how to think like an analyst: to search for hidden leverage in the problem’s assumptions and exploit them.
Moving forward, I'm now studying from John and Barbara Hubbard’s Vector Calculus, Linear Algebra, and Differential Forms. But I carry with me the conviction that I can learn real mathematics independently, as long as I'm willing to wrestle with the ideas. If you're hesitating on whether to pick up Abbott's Understanding Analysis, my advice would be to brush up on proof techniques, and dive in. It will be difficult, but finishing it might just make you fall in love with analysis.