Real analysis Study Guide

📖 Core Concepts Real numbers: the unique complete ordered field; every non‑empty set bounded above has a least upper bound (supremum). Metric on ℝ: distance given by \(d(x,y)=|x-y|\); the metric topology coincides with the order topology. Sequences Convergence: \(\forall\varepsilon>0\;\exists N\) such that \(n\ge N\Rightarrow|an-L|<\varepsilon\). Cauchy: \(\forall\varepsilon>0\;\exists N\) such that \(m,n\ge N\Rightarrow|am-an|<\varepsilon\). In ℝ, Cauchy ⇔ convergent (completeness). Monotone + bounded ⇒ convergent. Function sequences Pointwise convergence: the ε–N condition holds for each fixed x. Uniform convergence: the same N works simultaneously for all x in the domain. Uniform convergence preserves continuity, integrability, and differentiability. Compactness (ℝⁿ): a set is compact iff it is closed and bounded (Heine–Borel). Equivalent formulations: finite subcover property, every sequence has a convergent subsequence whose limit stays in the set. Continuity Pointwise: \(\forall\varepsilon>0\;\exists\delta>0\) s.t. \(|x-c|<\delta\Rightarrow|f(x)-f(c)|<\varepsilon\). Uniform: the same δ works for all points of the set. On a compact set, continuous ⇒ uniformly continuous. Absolute continuity: stronger; controls total variation over collections of small intervals. Differentiation Derivative: \(f'(c)=\displaystyle\lim{h\to0}\frac{f(c+h)-f(c)}{h}\). Differentiable ⇒ continuous. Function classes: \(C^{k}\) (k‑th derivative continuous), \(C^{\infty}\) (smooth), analytic (locally equal to a convergent power series). Series of real numbers Convergent ⇔ partial sums \(SN\) converge. Absolute convergence: \(\sum|an|\) converges ⇒ the original series converges. Conditional convergence: convergent but not absolutely. In ℝ, absolute ⇔ unconditional convergence. Key theorems (high‑yield): Bolzano–Weierstrass, Mean Value Theorem, Fundamental Theorem of Calculus, Dini’s theorem, Arzelà‑Ascoli, Stone‑Weierstrass, Banach fixed‑point, Taylor’s theorem, Heine–Borel. --- 📌 Must Remember Least Upper Bound Property → ℝ is complete → every Cauchy sequence converges. Monotone Convergence Theorem (sequences): monotone and bounded ⇒ convergent. Uniform convergence ⇒ limit function inherits continuity, integrability, differentiability. Heine–Borel: In ℝⁿ, compact ⇔ closed & bounded. Continuous on compact ⇒ uniformly continuous. Absolute convergence ⇒ convergence ⇒ unconditional convergence (in ℝ). Bolzano–Weierstrass: every bounded sequence in ℝ has a convergent subsequence. Mean Value Theorem: ∃ c ∈ (a,b) with \(f'(c)=\dfrac{f(b)-f(a)}{b-a}\). Fundamental Theorem of Calculus: \(\displaystyle\inta^b f(x)\,dx = F(b)-F(a)\) if \(F' = f\). Dini’s theorem: monotone sequence of continuous functions on a compact set that converges pointwise actually converges uniformly. --- 🔄 Key Processes Proving a sequence converges Identify a candidate limit \(L\). Show \(\forall\varepsilon>0\;\exists N\) with \(|an-L|<\varepsilon\) for \(n\ge N\). Shortcut: If the sequence is monotone and bounded, invoke the Monotone Convergence Theorem. Using the Cauchy criterion Verify \(\forall\varepsilon>0\;\exists N\) such that \(|am-an|<\varepsilon\) for all \(m,n\ge N\). In ℝ, this immediately yields convergence. Testing uniform convergence of \((fn)\) on \(E\) Find an \(N\) (independent of \(x\)) such that \(|fn(x)-f(x)|<\varepsilon\) for all \(x\in E\) when \(n\ge N\). Common tool: Weierstrass M‑test (if \(|fn(x)|\le Mn\) and \(\sum Mn\) converges). Checking compactness of a subset of ℝⁿ Verify closed (contains all limit points) and bounded (fits inside some ball). Alternatively, show every sequence in the set has a convergent subsequence (subsequential compactness). Applying the Mean Value Theorem Confirm \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\). Conclude existence of \(c\) with the slope equality. Evaluating a definite integral via FTC Find an antiderivative \(F\) of \(f\). Compute \(F(b)-F(a)\). Determining absolute vs conditional convergence Test \(\sum |an|\) (comparison, ratio, root tests). If \(\sum|an|\) diverges but \(\sum an\) converges, the series is conditional. --- 🔍 Key Comparisons Pointwise vs Uniform Convergence Pointwise: \(N\) may depend on the point \(x\). Uniform: a single \(N\) works for all \(x\) simultaneously. Absolute vs Conditional Convergence Absolute: \(\sum |an|\) converges ⇒ original series converges. Conditional: series converges but \(\sum |an|\) diverges. Continuous vs Uniformly Continuous Continuous: δ may depend on the point \(c\). Uniformly continuous: one δ works for every pair of points in the domain. Cauchy vs Convergent (in ℝ) Cauchy: terms become arbitrarily close to each other. Convergent: terms approach a specific limit. In ℝ they are equivalent; not true in incomplete spaces. Closed & Bounded vs Compact (ℝⁿ) In Euclidean space they are equivalent (Heine–Borel). Absolute vs Unconditional Convergence (ℝ) In ℝ they coincide; in general Banach spaces they differ. --- ⚠️ Common Misunderstandings “Every bounded monotone sequence converges” – true only when the sequence is both monotone and bounded. Uniform continuity follows from continuity – false on non‑compact domains (e.g., \(f(x)=x^2\) on ℝ). If \(\lim an = 0\) then \(\sum an\) converges – the necessary condition is not sufficient (harmonic series). Rearranging a convergent series never changes its sum – only true for absolutely (or unconditionally) convergent series. Differentiability ⇒ higher‑order differentiability – not; a function can be differentiable once but not have a continuous derivative. --- 🧠 Mental Models / Intuition Completeness: think of the real line as a solid rope with no missing points; any “Cauchy tug” eventually settles at a point on the rope. Uniform convergence: the whole family of graphs “locks together” after some index; you can pick a single N that works everywhere. Compactness: a set you can contain in a closed box; every sequence inside must have a “stay‑inside” limit point. Absolute convergence: the total “size” of the series is finite, so shuffling terms can’t change the sum. Cauchy sequence: “after a while, all terms are within a tiny cloud of each other.” --- 🚩 Exceptions & Edge Cases Monotone sequences: need boundedness to guarantee convergence. Uniform continuity may fail on open or unbounded intervals (e.g., \(f(x)=\frac{1}{x}\) on \((0,1]\)). Pointwise convergence does not preserve continuity; uniform convergence is required. Absolute ⇒ unconditional only in ℝ; in Banach spaces unconditional convergence can occur without absolute convergence. Differentiability does not imply the derivative is continuous (e.g., \(f(x)=x^2\sin(1/x)\) at \(0\)). --- 📍 When to Use Which Cauchy test: when you have no obvious limit but can bound \(|am-an|\). Monotone + bounded: quickest route to convergence for monotone sequences. Heine–Borel: decide compactness of a subset of ℝⁿ. Uniform convergence test (Weierstrass M‑test): when each \(fn\) is bounded by a summable sequence \(Mn\). Mean Value Theorem: to locate a point with a specific derivative value or to estimate differences. Fundamental Theorem of Calculus: evaluate definite integrals when an antiderivative is known. Absolute convergence test (ratio, root, comparison): first check for absolute convergence; if it fails, investigate conditional convergence. Dini’s theorem: when you have a monotone sequence of continuous functions on a compact set and already know pointwise convergence. Arzelà‑Ascoli: to prove pre‑compactness of a family of functions (e.g., for existence of convergent subsequences). --- 👀 Patterns to Recognize “Bounded + monotone ⇒ converge” appears in many sequence problems. ε‑N with “for all x ∈ E” signals uniform convergence or uniform continuity. Closed & bounded → compact repeatedly used for existence theorems (extreme value, uniform continuity). Series with alternating decreasing terms → suspect conditional convergence (Leibniz test). Presence of a supremum/infimum argument → often a completeness or least‑upper‑bound proof. “Every bounded sequence has a convergent subsequence” → Bolzano–Weierstrass; look for subsequence extraction. --- 🗂️ Exam Traps Choosing pointwise convergence when the question asks about continuity of the limit – you need uniform convergence. Assuming a continuous function on an open interval is uniformly continuous – false; must be on a compact interval. Neglecting to verify boundedness for a monotone sequence – an unbounded monotone sequence diverges to ±∞. Using the Ratio Test on a series with terms that are not positive – the test applies to absolute values; otherwise you may misclassify conditional convergence. Rearranging a conditionally convergent series – can change the sum; answer choices that claim invariance are traps. Applying the Mean Value Theorem without checking differentiability on the open interval – missing the hypothesis invalidates the conclusion. Confusing absolute convergence with unconditional convergence in a Banach‑space context – in ℝ they coincide, but the exam may present a Banach‑space scenario. ---