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Summary Real Analysis

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Real Analysis Notes and Exercises Description: This document provides comprehensive notes and exercises on key topics in Real Analysis, designed for undergraduate mathematics students. It spans three pages, covering fundamental concepts such as sequences and limits, series, and continuity. Each s...

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Uploaded on July 20, 2024
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Written in 2023/2024
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real analysis
mathematics
sequences
limits
series
series
convergence
continuity
convergence
continuity
theorems
exercises
examples
mathematics lessons
series
intermediate value theorem

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Real Analysis Notes and Exercises
Page 1: Sequences and Limits

Notes
1. Definitions:

 - Sequence: A sequence is a function from the set of natural numbers ℕ to a set S.
Typically, S ⊆ ℝ, and a sequence is written as (a_n)_{n=1}^∞ or simply (a_n).
 - Limit of a Sequence: A sequence (a_n) has a limit L ∈ ℝ if for every ε > 0, there exists
a natural number N such that |a_n - L| < ε for all n ≥ N.

2. Theorems:

 - Uniqueness of Limits: If a sequence (a_n) has a limit, then it is unique.
 - Squeeze Theorem: If a_n ≤ b_n ≤ c_n for all n ≥ N and lim_{n → ∞} a_n = lim_{n → ∞}
c_n = L, then lim_{n → ∞} b_n = L.

3. Examples:

 - Consider the sequence a_n = 1/n. We claim lim_{n → ∞} a_n = 0.
 - Given ε > 0, choose N = ⌈1/ε⌉. For n ≥ N, 1/n ≤ 1/N < ε.

Exercises
1. 1. Prove that lim_{n → ∞} (3n+2)/(2n+1) = 3/2.
Solution:
lim_{n → ∞} (3n+2)/(2n+1) = lim_{n → ∞} (3 + 2/n)/(2 + 1/n) = 3/2

2. 2. Show that the sequence a_n = (-1)^n does not have a limit.
Solution:
a_n = (-1)^n alternates between 1 and -1.
For the limit to exist, the sequence must converge to a single value, but here it does not.

3. 3. Determine whether the sequence a_n = √(n+1) - √n converges and find its limit if it
does.
Solution:
a_n = √(n+1) - √n = (√(n+1) - √n)(√(n+1) + √n)/(√(n+1) + √n) = 1/(√(n+1) + √n)
lim_{n → ∞} a_n = lim_{n → ∞} 1/(√(n+1) + √n) = 0

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