Let {$t_n$} = $\sum_{k=n}^{\infty}a_k$. Prove that the sequence {$t_n$} converges to 0. Suppose that $\sum_{k=1}^{\infty}a_k$ is a convergent series. 
a) Prove that the series $\sum_{k=n}^{\infty}a_k$ converges for each positive integer n.
b)Let {$t_n$} = $\sum_{k=n}^{\infty}a_k$. Prove that the sequence {$t_n$} converges to 0.
So I am done with part a and understand it. But I am having a hard time conceptualizing b. Would the terms of that series be decreasing? Any help would be appreciated. Having a hard time figuring it out. 

Since $\sum a_k$ is convergent the sequence $s_n=\sum_{k=1}^na_k$ of partial sums is convergent and hence Cauchy. This means that:
$$\forall \epsilon>0,\,\exists n_0 \in \mathbb{N},\,\forall m,n\geq n_0,\,|s_m-s_n|\leq \epsilon$$
Notice that if $m>n$, then $s_m-s_n =\sum_{k=n+1}^ma_n$. Letting $m\to\infty$, we concude that 
$$\forall \epsilon>0,\,\exists n_0 \in \mathbb{N},\,\forall n\geq n_0,\,\left|\sum_{k=n+1}^\infty a_n\right|\leq \epsilon.$$
Notice that this step is allowed because of part (a).
It suffices to note that the statement above means precisely that $t_n\to0$.
