Suppose that UCC is open and connected and a E U. Let F:= {f € H(U)| Re(f)> 0, f(a) 1} Show that Fis normal.
Let F be a field of characteristic p > 0. Show that f = t4 +1 € F[t] is not irreducible. Let K be a splitting field of f over F. Determine which finite field F must contain so that K = F.
Suppose that A1,A2,.., Ak are mutually exclusive events and P(B)>0. Prove that
PROVE: 4. If f : R → R is a strictly increasing function, f(0) = 0, a > 0 and b > 0, then
For all n E N prove that 0 <e- > < 2 k!“ (n + 1)! k=0 Hint: Think about Taylor approximations of the function e".
Suppose that A is diagonalizable and all eigenvalues of A are positive real numbers. Prove that det (A) > 0. (1 point) Suppose that A is diagonalizable and all eigenvalues of A are positive real numbers. Prove that det(A) > 0. Proof: , where the diagonal entries of the diagonal matrix D are Because A is diagonalizable, there is an invertible matrix P such that eigenvalues 11, 12,...,n of A. Since = det(A), and 11 > 0,..., n > 0,...
4. (a) Suppose that limz-c f(x) = L > 0. Prove that there exists a 8 >0 such that if 0 < 12 – c < 8, then f(x) > 0. (b) Use Part (a) and the Heine-Borel Theorem to prove that if is continuous on (a, b) and f(x) > 0 for all x € (a, b), then there exists an e > 0 such that f(x) > € for all x € [a, b].
Prove or Disprove that: If a > 0 and b are two rational numbers, then a' is a rational number.
3. Let X be a random variable and denote by Mx(t) its MGF. Prove that, for any t > 0, we have P[X >Mx(t)e
real analysis hint 13 Suppose fis a continuous function on R', with period 1. Prove that lim Σ f(a)-| f(t) dt 0 for every irrational real number α. Hint: Do it first for f(t)= exp (2nikt), k = 0,±1, ±2, 4.13 Let 2 be the set of functions of form P(t)-Σ_NQC2nikt. The equality holds for functions in . For given ε > 0, there is a P E 2 such that llf-Plloo < ε. Then