You're working on proving that if $f(n) = \theta(g(n))$ and $g(n) = \theta(h(n))$, then $f(n) = \theta(h(n))$. This is a transitivity property of Big-$\Theta$ notation. Let’s break down the proof clearly step by step.

Step-by-Step Proof:

1. Start with the definition of Big-$\Theta$ for $f(n) = \theta(g(n))$: By definition, we know that: $\exists \ c_1, c_2, n_0 \ \ \text{such that} \ \ \forall n \ge n_0, \ \ c_1 \cdot g(n) \leq f(n) \leq c_2 \cdot g(n)$ This means that $g(n)$ bounds $f(n)$ from both below and above for all $n \ge n_0$.

2. Now, consider $g(n) = \theta(h(n))$: Similarly, by the definition of Big-$\Theta$, we know: $\exists \ d_1, d_2, n_1 \ \ \text{such that} \ \ \forall n \ge n_1, \ \ d_1 \cdot h(n) \leq g(n) \leq d_2 \cdot h(n)$ This means that $h(n)$ bounds $g(n)$ from both below and above for all $n \ge n_1$.

3. Combine the inequalities: To prove $f(n) = \theta(h(n))$, we need to show that $h(n)$ bounds $f(n)$ from both below and above. This can be done by combining the two inequalities.

   • From $f(n) = \theta(g(n))$, we know: $c_1 \cdot g(n) \leq f(n) \leq c_2 \cdot g(n)$

   • From $g(n) = \theta(h(n))$, we know: $d_1 \cdot h(n) \leq g(n) \leq d_2 \cdot h(n)$

4. Substitute bounds for $g(n)$ into the inequality for $f(n)$: By substituting the bounds on $g(n)$ from the second inequality into the first inequality, we get: $c_1 \cdot (d_1 \cdot h(n)) \leq f(n) \leq c_2 \cdot (d_2 \cdot h(n))$ Simplifying: $(c_1 \cdot d_1) \cdot h(n) \leq f(n) \leq (c_2 \cdot d_2) \cdot h(n)$

5. Conclusion: The final inequality shows that there exist constants $e_1 = c_1 \cdot d_1$ and $e_2 = c_2 \cdot d_2$ such that: $e_1 \cdot h(n) \leq f(n) \leq e_2 \cdot h(n)$ This is exactly the definition of $f(n) = \theta(h(n))$.

Conclusion:

Thus, we've shown that if $f(n) = \theta(g(n))$ and $g(n) = \theta(h(n))$, then $f(n) = \theta(h(n))$. This proves the transitivity property of Big-$\Theta$ notation.

Would you like any further details or clarifications?

Here are 5 related questions:

1. How does Big-$O$ notation relate to Big-$\Theta$?
2. Can you prove a similar result for Big-$O$ or Big-$\Omega$ notation?
3. What are some examples of functions where $f(n) = \theta(g(n))$?
4. How does the transitivity property help in algorithm analysis?
5. What are the limitations of Big-$\Theta$ notation in asymptotic analysis?

Tip:

When proving relationships with asymptotic notations, always carefully track how constants combine during substitutions to maintain the correct bounds.