Metric Space

See atatched

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 g ...continues

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.

Real Analysis -- Real Numbers/Integers Theory

Let x be a real number, and let N be an integer ≥ 2. Prove that there exist integers P and Q such that: 1 ≤ q ≤ N and absolute value of [x-(P/Q)] < 1/(QN)

Covers, Series Convergence

Say the only tool given to you is a calculator which performs addition, subtraction, multiplication, and division. Let X=(see attached). Explain a practical way of computing X within an error of 10^8. Roughly how big is X?

Inner Product Space Convergence

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Show that 2 sequences are orthonormal

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Hilbert space/subspace

Please see the attached file for full problem description.