MTG Salvation

Old question. I learned a while back that: When a 4/4 trampler with double strike is blocked by a 2/2, the 2/2 died and opponent would take two damage, but the 4/4's second strike won't happen since the creature is dead.

Is it true or false?

That is false. Wherever you learned that from may have been confusing the situation with one where the attacker does not have trample. If the attacker does not have trample, the 2/2 will die and there will be nothing to deal the "second strike" damage to so it simply isn't dealt. Adding trample allows you to trample over <nothing> to the player.

False. Creatures with double strike deal damage in both damage steps. In terms of the regular combat damage step for these creatures, a creature being destroyed with first strike damage is exactly identical to a blocker getting hit by removal prior to damage. A creature without trample will not have anything to deal damage to, while a creature with trample (or a Thorn Elemental-style ability) will get to deal all of its damage to the defending player or planewalker. In your scenario, the defending player would take a total of 6 damage, 2 from the first combat damage step and 4 from the second.

It's possible you conflated double strike and trample with how creature with double strike and lifelink work, for which whether or not the blocker dies in the first damage step is more important.

Thank you both for the answer.