HONORS EXAM 2014 REAL ANALYSIS
Real analysis I
1. A sequence of non-negative real numbers a1, a2, . . . is called subadditive if am+n ≤ am + an for all m, n ≥ 1. Show that for any subadditive sequence, limn→∞ an/n exists and equals infn→∞ 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 dH on K(X) as follows: for A, B ∈ K(X), dH(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 {fn: [0, 1] → R: n = 1, 2, . . .} be a sequence of continuous functions. Prove or disprove: The functions {fn} converge to a function f uniformly on [0, 1] if and only if the corresponding graphs, {(x, fn(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 sn+1 = ½(sn + (2/sn)) 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 fn(x) = √nxn(1 - x). Discuss the convergence of {fn}, {f'n}, and { 0∫1fn(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.