The issue is with attempting to prove what the title mentions. I am told that the first part of my proof makes sense:
PROOF
An unsatisfiable sentence is false in all of its interpretations
Suppose A is an unsatisfiable sentence and B is any sentence
so, A is false in all of its interpretations
There is no interpretation in which A is true and B is false
A implies B
Now the issue I am having is continuing this proof and refining the second part to it (below) in order to make it make sense:
PROOF cont.
Now suppose B implies A
There is no interpretation where B is true and A is false
For all interpretations where A is false, B is false
A is false in all of its interpretations
so, B is false
so, Any sentence B implies an unsatisfiable sentence A
END PROOF
So the frist direction of the "if and only if" statement is done i.e. the direction where you assume that $A$ is not satisfiable. So for the second direction, why do you assume that B implies A? Especially, you can not here assume that A is not satisfiable (as this is what you want to prove) which you actually do on row 4 in the second part.
Instead for the second direction, assume that a sentence $A$ implies all other sentences. So especially it implies $\neg A$. Thus $A\rightarrow \neg A$ hold. However if $A$ is ever interpreted as true in any evaluation, this implication does not hold. Thus we conclude that $A$ allways needs to be interpreted as false i.e. $A$ is unsatisfiable.