(related to Proposition: Convergent Complex Sequences Are Bounded)
Let \((a_n)_{n\in\mathbb N}\) be a convergent complex sequence with \(\lim_{n\rightarrow\infty} a_n=a\). It follows that there is an \(N\in\mathbb N\) such that
\[|a_n - a| < 1\quad\quad\forall n\ge N.\] By virtue of the triangle inequality, we get:
\[|a_n| = |a_n - a + a|\le |a_n - a| + |a| < 1 + |a|\quad\quad\forall n\ge N.\]
Set \(B:=\max(|a_0|,|a_1|,\ldots,|a_{N-1}|,1 + |a|)\). With this positive constant real number, we have
\[|a_n| \le B\quad\quad\forall n\in\mathbb N.\]
Therefore, \((a_n)_{n\in\mathbb N}\) is bounded.
