# Prove that if $f(n) ∈ Θ(g(n))$ and $g(n) ∈ Θ(h(n))$ then $f(n) ∈ Θ(h(n))$.

I know that by the assumption that $f ∈ Θ(g)$ we know that there exist constants $c_0$ and $c_1$ in $\mathbb R^+$, and there exists $n_0 ∈ \mathbb N$ such that:

$$c_0 \cdot g(n)\le f(n)\le c_1 \cdot g(n),$$

but this is then where I get stuck, can anyone help me out?

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can anyone help me out? Yes I can: continue by writing down what $g\in\Theta(h)$ means, like you did for $f\in\Theta(g)$, and the light will come. –  Did Oct 14 '12 at 10:55

$$c_2 g(n) \leq f(n) \leq c_1 g(n)\\ c_4h(n) \leq g(n) \leq c_3 h(n)$$
Now plug in the bounds for $g(n)$ in the first inequality: $$c_2 c_4 h(n) \leq f(n) \leq c_1 c_3 h(n)$$ Since $c1 \cdot c3=c'$ and $c2 \cdot c4=c''$ are both constants: $$f(n)=\Theta (h(n))$$