Prove assertion that $3^{2n+5}$inO$(9^n)$ is true by definition. Clearly provide and demarcate the following elements in your proof:

$1.$ Give a formal definition of the asymptotic notation that includes the relevant inequality.

$2.$ State what the function $f(n)$ and $g(n)$ in the definition are with respect to the claim.

$3.$ State the goals of the proof; that is$,$ state what you would like to find$/$show $-$ the relevant multiplicative constant$($s$)$ and integer constant for which the inequality in the definition is true in some positive infinite interval.

$4.$ Provide detailed algebraic manipulation and mathematical arguments to derive the constants.

$5.$ Give concluding remarks that include how the constants that you derive make the inequality in the definition hold and the proposition true.