(i) Consider the sequence {an} on [0, 1] defined as follows: a = 0, a2 =...

   

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(i) Consider the sequence {an} on [0, 1] defined as follows: a = 0, a2 = 1, a3 = 1/2, a4 = 1/4, a5 = 3/4, a6 = 1/8, a7 = 3/8, as = 5/8, ag = 7/8,... What are the limit points of {an}, limsup and liminf? (ii) Consider the sequence {an} on [0, 1] defined as follows: a = 0, a2 = 1, a3 = 0, a = 1/2, a5 = a4 1, a6 = 0, a7 = 1/4, as = 2/4, ag = 3/4,a10 = 1, a11 = 0, a12 = 1/8, a13 = 2/8, a14 = 3/8, a15 = - 4/8, a16 = 5/8, a17 = 6/8, a18 = 7/8, a19 = 1, ... What are the limit points of {an}, limsup and liminf? (iii) For any functions f(x) and g(x) defined on (a, ) we have limsup (f(x) + g(x)) limsuproof(x) + liminfg(x) > liminf(f(x) + g(x)). (iv) Let f(x) and g(x) be any functions defined on (a, ) and suppose that liminf(g(x) = f(x)) 0. - Show that limsupf(x) limsuprg(x), and liminf f (x) < liminfg(x). (v) Give an example of two sequences {an} and {bn} such that liminf an + liminf bn < liminf (an + bn) < liminf an + limsup bn n n n n n < limsup(an + bn) < limsup an + limsup bn (i) Consider the sequence {an} on [0, 1] defined as follows: a = 0, a2 = 1, a3 = 1/2, a4 = 1/4, a5 = 3/4, a6 = 1/8, a7 = 3/8, as = 5/8, ag = 7/8,... What are the limit points of {an}, limsup and liminf? (ii) Consider the sequence {an} on [0, 1] defined as follows: a = 0, a2 = 1, a3 = 0, a = 1/2, a5 = a4 1, a6 = 0, a7 = 1/4, as = 2/4, ag = 3/4,a10 = 1, a11 = 0, a12 = 1/8, a13 = 2/8, a14 = 3/8, a15 = - 4/8, a16 = 5/8, a17 = 6/8, a18 = 7/8, a19 = 1, ... What are the limit points of {an}, limsup and liminf? (iii) For any functions f(x) and g(x) defined on (a, ) we have limsup (f(x) + g(x)) limsuproof(x) + liminfg(x) > liminf(f(x) + g(x)). (iv) Let f(x) and g(x) be any functions defined on (a, ) and suppose that liminf(g(x) = f(x)) 0. - Show that limsupf(x) limsuprg(x), and liminf f (x) < liminfg(x). (v) Give an example of two sequences {an} and {bn} such that liminf an + liminf bn < liminf (an + bn) < liminf an + limsup bn n n n n n < limsup(an + bn) < limsup an + limsup bn