I am also a beginner in Algorithm analysis, so this answer might need verification by some elder :P

start like this

$n \ge 10^{10}$, $\forall n \ge10^{10}$
raising exponent n on both side of inequality, we get
$\implies n^n\ge10^{10n}, \forall n \ge10^{10}$
$\implies 10^{10n}\le n^n,\forall n\ge 10^{10}$
$\implies 10^{10n}=O(n^n)$
Here the $c=1$ and $n_0=10^{10}$

The value of $c$ and $n_0$ must come from the derivation.

start like this

$n \ge 10^{10}$, $\forall n \ge10^{10}$
raising exponent n on both side of inequality, we get
$\implies n^n\ge10^{10n}, \forall n \ge10^{10}$
$\implies 10^{10n}\le n^n,\forall n\ge 10^{10}$
$\implies 10^{10n}=O(n^n)$
$\implies n^n \notin o(10^{10n})$
Here the $c=1$ and $n_0=10^{10}$

The value of $c$ and $n_0$ must come from the derivation.