> [!NOTE] Definiton > Let $f,g$ be real sequences then $g$ is big theta of $f,$ denoted $g=\Theta(f(n)),$ iff $g$ is [[Big O Relation on Real Sequences|big O]] and [[Big Theta Relation on Real Sequence|big Theta]] of $f$: that is there exist $c_{1},c_{2}>0$ and $N\in\mathbb{N}$ so that for all $n\geq N,$ $c_{1}|f(n)|\leq |g(n)|\leq c_{2} |f(n)|.$ # Properties By [[Big O of Real Sequence in Terms of Limit]], if $\frac{g(n)}{f(n)}$ tends to a non-zero real number as $n\to \infty$ then $g(n)$ is big theta of $f(n).$ By [[Big Theta gives Equivalence Relation on Real Sequences]], $f(n)=\Theta(g(n))$ and $g(n)=\Theta(h(n))$ implies $f(n)=\Theta(h(n)).$