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Let C he the curve [cee[1’rt2} + tweet“—tll — ein[1’ttf2]} fur Be at and let 2 [391+ms[x+y})i—cas{x+y}j. Find] F- 11115. (Hint: recall thatn= (d5,— —%}.}c Evaluate the line httegrali lxy :11: + 1:2 11y where C’ 15 the curve 1:111 the cardiad1*: 2 + cos B traveled counterclockwise

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 Let C_b(R) denote the space of all bounded, continuous functions f : R → C. Let C_0(R) denote the set of continuous functions f : R → C for which lim x→±∞ f(x) = 0. How do you prove that C_b(R) and C_0(R) are complete in the uniform metric?

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2. Let Dn be the dihedral group of symmetries of a regular n-gon, and let Cn ⊂ Dn be the subgroup of rotations. Let also H ⊂ Cn be an arbitrary subgroup. Prove that H is a normal subgroup of Dn.

Remark: H is a subgroup of Cn, so it is also a subgroup of Dn. Note that H is a normal subgroup of Cn, because Cn is Abelian. This means that ghg−1 ∈ H for any g ∈ Cn. What you need to prove is a stronger result: H is normal in Dn, meaning that ghg−1 ∈ H for any g ∈ Dn, and not only for any g ∈ Cn.

Hint: One possible approach is to show that for any element g ∈ Dn, the set gHg^-1 = {ghg^-1 | h ∈ H} is a subgroup of Cn. What is the order of this subgroup?

What do we know about subgroups of cyclic groups? Another approach is to use geometric arguments to prove that for any reflection g ∈ Dn Cn and any rotation r ∈ Cn, one has grg^-1= r^-1.

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Please help with the attached question. The question is a fill-in-the blank proof

Let E CR and F C R. Let X = {(r,y)|:r,yE IR and y = I}. Prove that ifX C E X F, thenE x F = R2 by filing out gaps in the following argument. Proof. We are given that X C _. We need to show that _ and Let (11,5) E E X F. Then, a E_ and b E_. If a. E_, then a E_ because E C_. If I; E_Tthen b E_ because F C_. Therefore, (o,b) E 13.2 because _. We showed that _. Let ((1,5) E 1R2. Then, a E_ and b E_. Therefore, by definition of X , we have (and) E Kand (b,b) E X. Since (the) E X and _, we have _. Since (b,b) E X and we have _.Therefore, (a, b) E E X F. We showed that _. _T

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