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Proofs : Bounded Sequences

Suppose the sequences {a_n}_n and {b_n}_n are both bounded above.

a) Prove that for all n in the naturals sup{a_k + b_k: k>/=n} is less than or equals sup{a_k:k>/=n} + sup{b_k:k>/=n}

b) Use this to conclude: limsup (a_n + b_n) is less than or equals limsup(a_n) + limsup(b_n)

(all limits are n--> infinity)

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Solution Summary

Bounded sequences are investigated.