Topics in Real Analysis
1. Foundations of Real Numbers
(a) Completeness Property
The real number system \( \mathbb{R} \) is complete, meaning every non-empty subset of \( \mathbb{R} \) that is bounded above has a least upper bound (supremum).
Example: The set \( S = \{ x \in \mathbb{R} \mid 0 \leq x < 1 \} \) is bounded above by 1. The supremum is \( \sup S = 1 \).
(b) Archimedean Property
The Archimedean property states that for any real numbers \( x > 0 \) and \( y \), there exists an integer \( n \) such that \( n \cdot x > y \).
Example: For \( x = 0.1 \) and \( y = 1000 \), we can find an integer \( n = 10001 \) such that \( n \cdot x = 1000.1 > 1000 \).
(c) Countable and Uncountable Sets
\( \mathbb{Q} \) is countable because we can list its elements in a sequence, while \( \mathbb{R} \) is uncountable due to the diagonalization argument.
Example: Cantor's diagonal argument shows \( \mathbb{R} \) is uncountable by constructing a real number not in any given enumeration of real numbers.
2. Sequences
(a) Convergence
A sequence \( (a_n) \) converges to \( L \) if for every \( \varepsilon > 0 \), there exists \( N \in \mathbb{N} \) such that for all \( n > N \), \( |a_n - L| < \varepsilon \).
Example: \( a_n = \frac{1}{n} \) converges to 0 because for any \( \varepsilon > 0 \), choosing \( N > \frac{1}{\varepsilon} \) ensures \( |a_n - 0| < \varepsilon \) for \( n > N \).
(b) Bolzano-Weierstrass Theorem
Every bounded sequence has a convergent subsequence.
Example: The sequence \( a_n = (-1)^n \) is bounded but oscillatory. Its subsequence \( a_{2n} = 1 \) converges to 1.
3. Series
(a) Convergence Tests
A series \( \sum_{n=1}^\infty a_n \) converges if the terms \( a_n \to 0 \) and satisfies specific criteria such as the Ratio Test.
Example: The geometric series \( \sum_{n=0}^\infty r^n \) converges to \( \frac{1}{1-r} \) for \( |r| < 1 \).
4. Limits and Continuity
(a) Formal Definition of Limit
\( \lim_{x \to c} f(x) = L \) if for every \( \varepsilon > 0 \), there exists \( \delta > 0 \) such that \( |x - c| < \delta \) implies \( |f(x) - L| < \varepsilon \).
Example: Prove \( \lim_{x \to 2} (3x + 1) = 7 \): For \( \varepsilon > 0 \), let \( \delta = \frac{\varepsilon}{3} \). Then \( |x - 2| < \delta \) ensures \( |(3x + 1) - 7| < \varepsilon \).
(b) Intermediate Value Theorem
If \( f \) is continuous on \([a, b]\) and \( f(a) \neq f(b) \), then \( f(c) = k \) for some \( c \in (a, b) \) whenever \( k \) lies between \( f(a) \) and \( f(b) \).
Example: \( f(x) = x^3 - x - 2 \) has a root in \( [1, 2] \) since \( f(1) = -2 \) and \( f(2) = 4 \).
5. Differentiation
(a) Mean Value Theorem
If \( f \) is continuous on \([a, b]\) and differentiable on \((a, b)\), there exists \( c \in (a, b) \) such that \( f'(c) = \frac{f(b) - f(a)}{b - a} \).
Example: For \( f(x) = x^2 \) on \([1, 3]\), \( f'(c) = 2c = \frac{3^2 - 1^2}{3 - 1} = 4 \). Thus, \( c = 2 \).
6. Integration
(a) Riemann Integrability
A function is Riemann integrable if its upper and lower sums converge to the same value as the partition is refined.
Example: The function \( f(x) = x \) is Riemann integrable on \([0, 1]\) with \( \int_0^1 x \, dx = \frac{1}{2} \).
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