HONORS EXAM 2014 REAL ANALYSIS

Real analysis I

1. A sequence of non-negative real numbers a₁, a₂, . . . is called subadditive if a_m+n ≤ a_m + a_n for all m, n ≥ 1. Show that for any subadditive sequence, lim_n→∞ a_n/n exists and equals inf_n→∞ an/n.

2. Let (X, d) be a metric space, and let K(X) be the space of all nonempty compact subsets of X. We define the Hausdorff metric d_H on K(X) as follows: for A, B ∈ K(X), d_H(A, B) is the smallest ε such that for every point a in A, there exists a point b in B with d(a, b) ≤ ε, and for every point b in B, there exists a point a in A with d(a, b) ≤ ε.

(a) Let S be the set of closed intervals in R, that is, the set {[x, y]: x ≤ y}. Is S open in K(R) with the Hausdorff metric? Closed? Neither?

(b) Given a set Y in the space X, its boundary, bdY, is the set bdY = cl(Y) ∩ cl(X\Y ). Show that the map ∂: K(R) → K(R) given by ∂(Y) = bdY is well defined, and determine whether it is continuous under the Hausdorff metric.

(c) Let {f_n: [0, 1] → R: n = 1, 2, . . .} be a sequence of continuous functions. Prove or disprove: The functions {f_n} converge to a function f uniformly on [0, 1] if and only if the corresponding graphs, {(x, f_n(x)) ∈ R²: x ∈ [0, 1]}, converge to the graph of f in K(R²) with the Hausdorff metric.

3. Recall the Intermediate Value Theorem:

Let f: [a, b] → R be a continuous function, and y any number between f(a) and f(b) inclusive. Then there exists a point c ∈ [a, b] with f(c) = y.

(a) Prove the Intermediate Value Theorem.

(b) Prove or disprove the following fixed point theorem:

Let g: R → R be a continuous function, and x₁ and x₂ distinct points such that g(x₁) = x₂ and g(x₂) = x₁. Then there exists a fixed point x (that is, a point x such that g(x) = x).

4. (a) Prove the following attracting fixed point theorem. Let f: R → R be a twice-differentiable function, and let x₀ be a point such that f(x₀) = x₀ and |f'(x₀)| < 1. Then x₀ is attracting, that is, there is an interval I containing x₀ in its interior such that f(I) ⊂ I and the sequence x, f(x), f(f(x)), . . . converges to x₀ for all x in I.

(b) Show that the sequence defined by s₁ = 1 and s_n+1 = ½(s_n + (2/s_n)) for n = 1, 2, . . . converges to √2. (This is the Babylonian method for computing square roots.)

5. Define a sequence of functions f₁, f₂, . . . : [0, 1] → R by f_n(x) = √nxⁿ(1 - x). Discuss the convergence of {f_n}, {f'_n}, and { ₀∫¹f_n(x) dx} as n → ∞.

Real analysis II

6. Prove that for every n × n matrix A sufficiently near the identity matrix, there is a square-root matrix B (i.e., a solution to B² = A). Show that the solution is unique if B must also be sufficiently near the identity matrix.

7. Let T ⊂ R³ be the torus (2 - √(x² + y²))² + z² = 1, and let ω be the 2-form, defined on R³\{0^→}, given by

ω = (x dy ∧ dz - y dx ∧ dz + z dx ∧ dy/(x² + y² + z²)^3/2).

(a) Show that ω is closed.

(b) Compute ∫_Tω.

(c) Compute ∫_S^2 ω, where S² is the unit sphere in R³.

8. For what values of c will the set {(x, y, z): x³ + y³ + z³ - 2xyz = c} be a 2-manifold?

9. (a) Compute ∫∫∫_R^3 f(2x, 3y, 4z) dx dy dz, given that ∫∫∫_R^3 f(x, y, z) dx dy dz = 1.

(b) Define S ⊂ R² to be the set {(x, y): -1 ≤ x ≤ 1, 0 ≤ y ≤ 2 √(1 - |x|)}. Compute

∫∫_s (√2/2) √(√(x²+y²)+x)dxdy

Hint: The function g(u, v) = (u² - v², 2uv) maps the unit square {(u, v) : 0 ≤ u ≤ 1, 0 ≤ v ≤ 1} one-to-one onto S.