Metric Space

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Real Analysis - Bounded Continuity/Differentiability

I am currently not understanding the subject material. If you could give me a full, thorough answer ("proved rigorously" not just simple statements) that I would be able to go through step by step and see your logic, it would be greatly appreciated. Problem: Let f: [0, ∞) → R be a bounded fuction. For all X greater than or equal to 0, let G(x)=sup{f(t): 0 is less than or equal to t is less than or equal to x} a) Show that if f is continuous, g is also continuous. Is the converse also true? Justify. b) If f is differentiable and continuous, is g also differentiable and continuous? Justify. Thank you so much!

Real Single-Variable Analysis Problem

Let x be an irrational number. Prove that there exist infinitely many fractions (p/q) with p and q as integers such that: abs(x-[p/q]) < 1/(q^2)

Real Analysis - Short Problem, involving series I believe

Say the only tool you have available to you is a pocket calculator which performs addition, subtraction, multiplication, and division, accurate to 15 decimal places. Explain a practical way to compute integral from 0 to 1 of e^[-(x^2)] to within an error less than 10^-8. Prove that the method works.

Real Analysis Problem

Prove rigorously: Let N be an integer > or equal to 2, and let Xsub0....Xsubn E [0,1). Prove that there exist i and j with i not equal to j such that abs (xsubi-xsubj) < 1/n.