In this note, we will show the necessary conditions for two sequences of reals, \(a_k\) and \(b_k\) such that equality is reached by the Cauchy-Schwarz Inequality. I believe this fact is already presented in the Cauchy-Schwarz Brilliant Wiki, but I don't believe it was shown in this manner. First, we begin with a statement of the inequality:

\[\left(\sum_{k=1}^{n} a_kb_k\right)^2 \leq \left(\sum_{k=1}^{n} a_k^{2}\right)\left(\sum_{k=1}^{n} b_k^{2}\right)\]

Now, subtracting to yield a non-negative result,

\[\left(\sum_{k=1}^{n} a_k^{2}\right)\left(\sum_{k=1}^{n} b_k^{2}\right) - \left(\sum_{k=1}^{n} a_kb_k\right)^2 \geq 0\]

This expression can be expanded and then neatly, intuitively compacted into summation notation. Also, we'll be leaving behind the inequality sign in exchange for an equal sign, since we're focusing on the equality case.

\[(a_1^2+a_2^2+...+a_n^2)(b_1^2+b_2^2+...+b_n^2) - (a_1b_1 + a_2b_2 + ... + a_nb_n)^2 = 0\]

\[\left(\sum_{i=1}^{n}\sum_{j=1}^{n} a_i^{2}b_j^{2}\right) - \left(\sum_{i=1}^{n}\sum_{j=1}^{n} a_ib_ia_jb_j \right) = 0\]

\[\frac{1}{2}\left(\sum_{i=1}^{n}\sum_{j=1}^{n} a_i^{2}b_j^{2} + a_j^2b_i^2\right) - \left(\sum_{i=1}^{n}\sum_{j=1}^{n} a_ib_ia_jb_j \right) = 0\]

Now, combining our double sums:

\[\frac{1}{2}\left(\sum_{i=1}^{n}\sum_{j=1}^{n} a_i^{2}b_j^{2} -2a_ib_ja_jb_i + a_j^2b_i^2\right) = \frac{1}{2}\left(\sum_{i=1}^{n}\sum_{j=1}^{n} (a_ib_j - a_jb_i)^2 \right) = 0\]

If \((b_1, b_2, ... , b_n) \neq 0\), then \(b_k \neq 0\) for some \(k\). For equality to occur, the above expression must equal \(0\), and so each term of the double sum must equal \(0\). Considering only the positive terms \(b_k\), it follows that equality holds with the Cauchy-Schwarz Inequality if and only if \(a_kb_i = a_ib_k\) for all \(i\) such that \(1 \leq i \leq n\) and for some value of \(k\).

Dividing both sides of this relation by \(b_i\) and \(b_k\) and setting \(\gamma = \frac{a_k}{b_k}\), we reach the fact that \(\frac{a_i}{b_i} = \gamma\). In other words, our sequences \(a_k\) and \(b_k\) must be \(\textit{proportional}\) for the equality case of the Cauchy-Schwarz Inequality.

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