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I'm doing a course which contains foundation work on convergence.

I was suprised to see the book I am using uses phrases such as....

" This sequence clearly doesn`t converge "

for sequences such as 2,0,2,0,2,0,2,0.....

I was expecting it to say something like " By theorem 4.5, this sequence doesn`t converge "

I wouldn`t feel comfortable writing " This sequence clearly doesn`t converge " if, in an exam, I got a question which said " Prove that 2,0,2,0,2,0 doesn`t converge ".

Can anyone point me to basic theorems on convergence which are used to tackle simple questions like this?