**Lemma** Suppose that there exists $N \in \mathbb{N}$ such that $0\leq a_{n} \leq b_{n}$ for every $n\geq N$, Then - if $\sum b_{n} < \infty$ then $\sum a_{n} < \infty$, - if $\sum a_{n} = \infty$ then $\sum b_{n} = \infty$ **Proof**: Note that $\sum_{n=1}^{\infty} a_{n} <\infty \iff \sum_{n=N}^{\infty} a_{n} < \infty$ since If we define the sequence $u_{n} = a_{n+N-1}$ So we can apply the [[Comparison Test for Series With Non-Negative Terms|comparison test]] on $\sum_{n=1}^{\infty}u_{n}$ to show that it converges. **Remark** See Application in show that [[Euler's Number]] is well-defined.