Chapter 11.2, Problem 50ES

To determine

(a)

To prove:

That for the real-valued function $f$ and $g$, if $f\left(n\right)\text{ is }\Omega \left(g\left(n\right)\right)$, then $cf\left(n\right)\text{ is }\Omega \left(g\left(n\right)\right)$ for any real number $c$.

To determine

(b)

To prove:

That for the real-valued function $f$ and $g$, if $f\left(n\right)\text{ is }O\left(g\left(n\right)\right)$, then $cf\left(n\right)\text{ is }O\left(g\left(n\right)\right)$ for any real number $c$.

To determine

(c)

To prove:

That for the real-valued function $f$ and $g$, if $f\left(n\right)\text{ is }\theta \left(g\left(n\right)\right)$, then $cf\left(n\right)\text{ is }\theta \left(g\left(n\right)\right)$ for any real number $c$.