Convergent series have terms and tails tending to zero

Let \((a_n)_{n\geq1}\) be a sequence in \(\CC\) such that \(\sum_{n=1}^\infty a_n\) converges. Then \(a_n\to0\) as \(n\to\infty\), and \[ \sum_{n=N}^\infty a_n \convergesto{N\to\infty} 0 . \]