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Hence lim sup(sn+tn)≤L+M+ 2ε, since lim sup(sn+tn) is theleastupperbound of the subsequential limits.Since this works for anyε >0, we must havelim sup(sn+tn)≤L+M,as desired.(b). Find an example to show that equality may not hold in part (a).Solution:Letsn= (-1)nandtn= (-1)n+1. Thensn+tn= 0 for alln, solim supn→∞(sn+tn) = limn→∞(sn+tn) = 0.However lim supsn= 1 and lim suptn= 1.5
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Math,lim,Limit of a function,Limit of a sequence,Limit superior and limit inferior,Sn,subsequence