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Let akand bkbe convergent sequences with akLand bkMas kand let cbe a constant. Prove the indicated basic limit rules from Theorem 7.11. You may wish to model your proofs on the proofs of the analogous statements from Section 1.5.

Prove that akbkLM.

Short Answer

Expert verified

Hence, the theorem is proved.

Step by step solution

01

Step 1. Given Information.

The objective is to prove thatakbkLM.

02

Step 2. Forming the equations.

We use the definition of convergence for the sequence akand bk.

The sequence akconverges to L.

For given ε>0, there exists a positive integer such that

ak-L<ε2mfor kN..........(1)

The sequence bkconverges to M.

For given ε>0, there exists a positive integer Psuch that

bk-M<ε2L+1for kP...........(2)

There exists a positive integer msuch that

bkmfor allk...........(3)

03

Step 3. Proving the theorem.

Choose a positive integer Rsuch that R=maxN,P

akbk-LM=ak-Lbk+bk-MLak-Lbk+bk-ML<ε2m×m+ε2L+1×LforkR<ε2+ε2=ε

Thus, for kR,akbk-(LM)<εand hence, akbkis convergent.

Therefore, the value islimkakbk=LM.

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