How are transactions invalidated by nodes when both parties are offline in a Lightning Network? Let's imagine Alice wants to send multiple transactions to Bob. Both Alice and Bob have open channels with Carol, not with each other:

Alice &lt;-&gt; Carol &lt;-&gt; Bob

So Alice first sends 0.01 BTC to Bob. She sets up the multi-sig tx and sends it off to Carol, which then also Bob sees. Then Alice makes another transaction to Bob of 0.02 BTC, and sets an earlier time-lock in order to invalidate the earlier 0.01 tx.

Now both Alice and Bob have seen these 2 transactions and they both go offline. They are sure that Alice has given Bob 0.02 BTC.

My question is: how do they know that Carol will commit to the blockchain (when the moment comes) the second transaction rather than the first? Couldn't Carol simply send the 0.01 tx to the blockchain rather than the 0.02 one?

I understand that Bob could come online and post it himself, and because of the timelock, he could do it before the other one, thus invalidating it.

But in the scenario where both Alice and Bob are offline do they have to trust Carol to behave honestly?
AFAIU, if you have open lightning channels, you cannot go offline. You must be online in order to monitor the blockchain for old commitments. If you see an old commitment transaction, you immediately broadcast the punishment transaction, but you cannot do this if you go offline. So if you have open lightning channels, you must be online and if you go offline, you are trusting that the other person in the channel will not attempt to defraud you.