As indicated by the inquiry,

P(n) which is valid for all n ≥ 4 however P(1), P(2) and P(3) are false

Let \[P\left( n \right)\text{ }be\text{ }2n\text{ }<\text{ }n!\]

Thus, the instances of the given assertions are,

\[P\left( 0 \right)\Rightarrow 20\text{ }<\text{ }0!\]

\[i.e\text{ }1\text{ }<\text{ }1\Rightarrow false\]

\[P\left( 1 \right)\Rightarrow 21\text{ }<\text{ }1!\]

\[i.e\text{ }2\text{ }<\text{ }1\Rightarrow false\]

\[P\left( 2 \right)\Rightarrow 22\text{ }<\text{ }2!\]

\[i.e\text{ }4\text{ }<\text{ }2\Rightarrow false\]

\[P\left( 3 \right)\Rightarrow 23\text{ }<\text{ }3!\]

\[i.e\text{ }8\text{ }<\text{ }6\Rightarrow false\]

\[P\left( 4 \right)\Rightarrow 24\text{ }<\text{ }4!\]

\[i.e\text{ }16\text{ }<\text{ }24\Rightarrow valid\]

\[P\left( 5 \right)\Rightarrow 25\text{ }<\text{ }5!\]

\[i.e\text{ }32\text{ }<\text{ }60\Rightarrow \] valid, and so forth