Exercise 1.12

Question

Exercise 12: If $z_1, \cdots, z_n \in \mathbb C$, prove that 
    $$ |z_1 + \cdots + z_n| \leq |z_1| + \cdots + |z_n| $$

ghostofgarborg · Accepted Answer

Note that by the triangle inequality, $|z_1 + z_2| \leq |z_1| + |z_2|$. Assume the statement holds for $n-1$. Then 
    $$ |z_1 + \cdots + z_{n-1} + z_n| \leq |z_1 + \cdots + z_{n-1}| + |z_n| \leq |z_1| + \cdots + |z_n| $$
    which establishes the claim by induction.

Amit Awasthi · Answer

\begin{proof}[Proof by induction]
      For $n = 1$, there is nothing to show since $|z_1| = |z_1|$. For $n = 2$, $|z_1 + z_2| \le |z_1| + |z_2|$, which is true by Theorem $1.33(e)$. Suppose the inequality holds for $n$. Then,
      \begin{multline*}
        |(z_1 + \cdots + z_n) + z_{n+1}|\
        \le |z_1 + \cdots + z_n| + |z_{n+1}|\
        \le |z_1| + \cdots + |z_n| + |z_{n+1}|,
      \end{multline*}
      where the last inequality holds by applying the inductive hypothesis to $|z_2 + \cdots + z_n|$, and the inequality at each iteration holds by Theorem $1.33(e)$.
\end{proof}

Exercise 1.12

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Exercise 1.12

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