Compactness implies sequential compactness

Question

I am studying topology using Serge Lang Real Analysis (second edition).In proposition 3.7 the author writes the following :"Compactness implies sequential compactness " and goes on to give a proof that I find convincing . I am confused because the internet gives me the answer that compact doesn't imply sequential compact and gives counter-examples ( that i find a bit difficult to handle ).Can someone explain to me what is going on here is Serge Lang imposing other conditions that i didn't pay attention to or this is a typing mistakes ?