Let (an), (bn) be two bounded sequences. a.Show that limn→∞⊃ max(an,bn)≥max(limn→∞⊃an,limn→∞⊃bn). b. Give an example of (an),(bn) such that the strict inequality holds.

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Let (an), (bn) be two bounded sequences.     a.Show that limn→∞⊃ max(an,bn)≥max(limn→∞⊃an,limn→∞⊃bn).     b. Give an example of (an),(bn) such that the strict inequality holds.

Hana Larsen · Accepted Answer

a)In view of (an) and (bn) are bounded sequences,limn→∞⊃max(an,bn)≥max(limn→∞⊃an,limn→∞⊃bn)We already know that lim sup is on the rise,⇒an≤max(an,bn)⇒lim⊃ max(an,bn)≥lim⊃ an (1)Likewise,⇒bn≤max(an,bn)⇒lim⊃ max(an,bn)≥lim⊃ bn (2)Thus,lim⊃ max(an,bn)≥max(lim⊃ an,lim⊃ bn)b)Let&#039;s substitute (an)=1n,(bn)=12nwe know that (an) and (bn) are bounded,⇒max(an,bn)=max(1n,12n)=max{(1,12,13,…),(12,14,…)}lim⊃ max(an,bn)=1 (3)However, limn→∞⊃(an)=0 and limn→∞⊃(bn)=0max(limn→∞⊃an,limn→∞⊃bn)=0 (4)Then by the equation (3) and equation (4):lim⊃ max(an,bn)&amp;gt;max(lim⊃ an,lim⊃ bn)Therefore the given limn→∞⊃ max(an,bn)≥max(limn→∞⊃an,limn→∞⊃bn) is showed and the example of (an),

Let (a_n), (b_n) be two bounded sequences. a.Show that \lim_{n

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