What is wrong with the proof? I try to prove the identity:
$$\sum_{k=0}^n \binom{2n+1}{2k}=2^{2n}$$
but have troubles with the last step:
$$\sum_{k=0}^n \binom{2n+1}{k}=\sum_{k=0}^n  \left( \binom{2n}{2k} + \binom{2n}{2k-1}\right) =\sum_{j=-1}^{2n}\binom{2n}{j}=2^{2n}$$
But should not $j$ start from $0$? Is something wrong?

Following the comment of Nicolas and the partial develop of Tyler:
$$\sum_{k=0}^n \binom{2n+1}{2k}=\binom{2n+1}{0}+\sum_{k=1}^n  \left( \binom{2n}{2k} + \binom{2n}{2k-1}\right) =1+\color{red}{\sum_{k=1}^{n}\binom{2n}{2k}}+\sum_{k=1}^n\binom{2n}{2k-1}$$
Observe that the marked term on red are the even coefficients, but $2k=0$, of a complete binomial sum of the kind
$$\sum_{k=0}^{2n}\binom{2n}{k}=2^{2n}=\sum_{k=0}^{n}\binom{2n}{2k}+\sum_{k=1}^{n}\binom{2n}{2k-1}=1+\sum_{k=1}^{n}\binom{2n}{2k}+\sum_{k=1}^n\binom{2n}{2k-1}$$
In other words, what remains to prove are the identities
$$\sum_{k=0\\k\text{ even}}^{2n}\binom{2n}{k}=\sum_{k=0}^{n}\binom{2n}{2k}$$
$$\sum_{k=0\\k\text{ odd}}^{2n}\binom{2n}{k}=\sum_{k=1}^{n}\binom{2n}{2k-1}$$
