I am new to this forum so please bear with me if I miss any information. I am currently taking advanced calculus using the textbook Introduction to Analysis (5th Ed) by Edward Gaughan. I am having a lot of trouble answering the following question:Give an example of a set of real number which has exactly two accumulation points.I am super confused about the concept of accumulation points at this point. I don't know if I am overthinking the concept or what, but I really need some help in knowing if my guess is correctS = {1/n : n∈N} U {1 + 1/n : n∈N},
Claim 0 is a limit point of S, 0 is not an element of SGiven epsilon > 0, find N ∈N such that 0< 1/N< epsilon (by Archimedean principle)1/N ∈ (0- epsilon, 0 + epsilon) intersection S with 1/n not equal to 0, therefore 0 is a limit point.
Claim 1 is a limit point of S, 1 is not an element of SGiven epsilon > 0, find N ∈N such that 1< 1 + 1/N< epsilon (by Archimedean principle)1 + 1/N ∈ (1- epsilon, 1 + epsilon) intersection S with 1+ 1/n not equal to 0, therefore 1 is a limit point.
I am really struggling, and I am unsure if this is correct or not.