Since when are natural deduction proofs allowed to terminate with undischarged assumptions?

Question

So far I thought natural deduction proofs are NOT allowed to terminate with undischarged assumptions. However this seems to be what's happening here:

When I translate this Gentzen style proof into a Fitch style proof I get this:

As clearly visible in the Fitch style proof the proof ends without discharging assumptions $[B]$ and $[A]$.

However I don't think the author of the Gentzen style proof made a mistake, which makes me question my understanding of things so far. Can you help me clear up where I am going wrong?

The screenshot of the Gentzen style proof is from this recording of a presentation on type theory.

Answer 1

The Gentzen proof does discharge the $B \land A$ with the $\rightarrow \ I$, but indeed does not discharge the $B$ and the $A$.

So, it proves (in a very clumsy way) the sequent $B, A \vdash A \land B$.

Translated into a Fitch style proof, it has $B$ and $A$ as premises, and $A \land B$ as the conclusion:

Gentzen proofs without any undischarged assumptions would be proofs or arguments without any premises, and thus of stand-alone valid sentences.