I presented a professor with the following proof:

Prove that the empty set is closed.

Proof :

By definition :

is closed <=> cl

<=> (

cl

=>

)

cl

is the closure of the empty set,and

B(x,r) is a ball of radius r round x

But ,by definition again

cl

<=> for all r>0 ,B(x,r)

.................................................. ..................1

But , B(x,r)

=> (B(x,r)

or

) <=>

(B(x,r)

=>

)

And using (1) we get :

Thus ,we have proved:

(

cl

=>

)

And the empty set is closed.

The professor did not accept the proof as correct .

Do you agree with him??