6. (20 pts) Prove that there is no homeomorphism of the closed interval (-1,1] of the...
Let ne Nj. Prove that n < 2(6(n)).
Let U ? Rmxn. Prove that if UTI-In, then n < m.
5. Prove that f(z) = (2+1/2) is a conformal map from the half-disc {z = x +iy : 2< 1, y >0} to the upper half-plane. (Hint: The equation f(z) = w reduces to the quadratic equation z2 + 2wz +1 = 0, which has two distinct roots in C whenever w # £1. This is certainly the case if WE H.
Find the minimum and the maximum values of |z2 + (p + 1)i| on the closed disc {z ∈ C : |z| ≤ q + 1} Find the minimum and the maximum values of 122 + (p + 1)i| on the closed disc {z € C: |Z| <q+1}.
IDY in < oo and lim - Yn < 0o. Prove that lim,+ 1. Let In > 0. Yn > 0 such that lim,- Yn) < lim,-- In lim,+ Yn: i tn < oo and lim yn < . Prove that lim. In 1. Let In 20, yn 0 such that lim Yn) < limn+In lim + Yr
7. Denoting, as usual, by (a, b) an "open interval", {x : a < x < b} and by [a,b] the corresponding "closed interval", {x : a < x < b} of real numbers, prove that supla, b) = sup a, b = b and inf(a, b) = inf[a,b- other words, does the proof work for any ordered field? a. Is completeness relevant:In
real analysis. questions Prove that if lima In = 0 and > M for some M >0 and in 10 > 0, then lima (ny) - Asume 30 = 2,2-20+ In+1 = In + Prove that this sequence has a limit and find the limit. Prove that lim = L with L < if and only if every subsequence limo n L. Suppose that the sequence {an) is increasing and the sequence {yn) is decreasing. Moreover, lim a n -...
let a,b > 0 . Prove that DI < Val
Solve the inequalities. Write the solution sets in interval notation if possible. 6 (a) <o y +1 6 (b) <0 y +1 (c) 20 y+1 (d) >0 y +1
6. (20 pts.) The plane y-0 separates region 1 (y 0), which is a dielectric materia with c, -3.5, from region 2 (y < 0), which is free space. If the electric flux density in region 1 is given by D,-15a, +22ay -20a, [nC/m'], find D..