Use Mathematical Induction to prove that $$^{2n}C_n < 2^{2n-2}$$ when $n\geq5$.
My attempt :
For $n=1$ , I got $LHS = 252 < RHS = 256$
Assumption : $$^{2k}C_k < 2^{2k-2}$$
For $n=k+1$,
$$^{2(k+1)-2}C_{k+1} = 2\times ^{2k+1}C_{k}$$
$$= 2\times \{^{2k}C_{k}+^{2k}C_{k-1}\}$$
$$ < 2\times \{2^{2k-2}+^{2k}C_{k-1}\}$$
how can I proceed from this step?