Dirichlet's convergence test

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questions re: Dirichlet's convergence

Should the second line be the summation from i=0 to i=n-1 and the summation from i=0 to i=m-2?

Also, in the last line, $\vert\sum_{i=m}^n a_ib_i\vert<\epsilon$ < infinity, therefore $\sum a_nb_n$ converges?

Thanks.

Re: questions re: Dirichlet's convergence

Agree with first comment; the sums are not correctly indexed.

Re the last line, it is correct, but you are using the Cauchy criterion: I recommend you quote it.