1: (an) and (bn) are bounded sequences: (a) prove that limsup(-an) = -liminf(an) (b) for any...

1: (a_n) and (b_n) are bounded sequences:

(a) prove that limsup(-a_n) = -liminf(a_n)

(b) for any c>0, prove that

limsup(ca_n) = climsup(a_n)

and

liminf(ca_n) = climinf(a_n)

(c) prove that

limsup(a_n+b_n) ≤ (limsup(a_n)) + (limsup(b_n))

and

liminf(a_n+b_n) ≥ (liminf(a_n)) + (liminf(b_n))

(d) If a_n and b_n are made of nonnegative terms, prove that

limsup(a_nb_n) ≤ (limsup(a_n)) x (limsup(b_n))

and

liminf(a_nb_n) ≥ (liminnf(a_n)) x (liminf(b_n))

(e) prove that

limsup(a_n+1) = limsup(a_n)

and

liminf(a_n+1) = liminf(a_n)

Expert Solution

Colby Messinger answered 16 hours ago

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1: (an) and (bn) are bounded sequences: (a) prove that limsup(-an) = -liminf(an) (b) for any...

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