Product of Big-O Estimates/Sequences
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Theorem
Let $\sequence {a_n}, \sequence {b_n}, \sequence {c_n}, \sequence {d_n}$ be sequences of real or complex numbers.
Let:
- $a_n = \map \OO {b_n}$
- $c_n = \map \OO {d_n}$
where $\OO$ denotes big-$\OO$ notation.
Then:
- $a_n c_n = \map \OO {b_n d_n}$
Proof
Since:
- $a_n = \map \OO {b_n}$
there exists a positive real number $C_1$ and natural number $N_1$ such that:
- $\size {a_n} \le C_1 \size {b_n}$
for all $n \ge N_1$.
Similarly, since:
- $c_n = \map \OO {d_n}$
there exists a positive real number $C_2$ and natural number $N_2$ such that:
- $\size {c_n} \le C_2 \size {d_n}$
for all $n \ge N_2$.
Let:
- $N = \max \set {N_1, N_2}$
Then, for $n \ge N$, we have:
\(\ds \size {a_n c_n}\) | \(=\) | \(\ds \size {a_n} \size {c_n}\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds C_1 \size {b_n} \size {c_n}\) | since $n \ge N_1$ | |||||||||||
\(\ds \) | \(\le\) | \(\ds C_1 C_2 \size {b_n} \size {d_n}\) | since $n \ge N_2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds C_1 C_2 \size {b_n d_n}\) |
giving:
- $a_n c_n = \map \OO {b_n d_n}$
$\blacksquare$