Is this a sufficient proof for "For all $k \in\Bbb R$, if $k$ is odd, then $4k + 7$ is odd Can I just say 4k is obviously even and even + odd is always odd, so odd?
Or is that too simple, am I missing something?

Yes, its simple like that. But if you are making a test or something like that you have to be sure that you can use the fact that "even+odd is odd" without proving it... otherwise just prove it! (Not so difficult)
