I am currently stuck on the following question:
$$\forall x, y \in N: y \in \Omega(x) \rightarrow y^3 \in \Omega(x^3)$$
$$ x^3: x^3(n) = x(n) * x(n) * x(n) $$ $$ y^3: y^3(n) = y(n) * y(n) * y(n) $$
My current thinking is to assume y $\in$ $\Omega$(x), use the definition to somehow show that ccc is c and therefore the definition of $\Omega$ holds true for $y^3$ as well. But I can't seem to put together a chain of equalities to show this.
Am I on right track? Can someone point me to a solved example similar to this please?
Since $y=\Omega(x)$, we have that there exists a $c$ such that for large $n$ $cy(n)<x(n)$. Thus $c^3y(n)^3<x(n)^3$. Let $k=c^3$. Then $ky^3<x^3$, so $y^3=\Omega (x)$.