I'm re-writing your post for readability. Please check that this is what you meant to say.

So I'm writing an assignment in which I have to define what is understood by a Hermitian operator. My teacher has told me to define it as:

where and are the th and th basis vectors, respectively. Using this I then have to prove the more general definition:

with and being arbitrary vectors.

I've tried to do so but I have not yet succeeded. What I've done is to say: take a Hermitian operator Since it's Hermitian it must satisfy

which, when dotted with an arbitrary ket and bra produces

Since is Hermitian this yields

Now the proof would work if but that's not right is it? Can anyone help me to prove this? I know it's simple, but I'm still finding it a little hard.

Is this what you meant?