Final answer:
The assertion is false; the sword given to Beowulf, Hrunting, is ineffective against Grendel's mother. It is a giant's sword found in her lair that ultimately allows Beowulf to defeat her.
Step-by-step explanation:
The statement that the sword given to Beowulf does a lot of damage to Grendel's mother is false. In the epic poem Beowulf, the hero does indeed receive a sword from Unferth, named Hrunting, which is renowned for its victories.
However, when Beowulf fights Grendel's mother in her underwater lair, this sword proves ineffectual against her tough skin. It's only when Beowulf finds a giant's sword in her lair, a weapon forged by giants and blessed with their magic, that he is able to defeat Grendel's mother by decapitating her. This indicates that the sword initially given to Beowulf, despite its fame, was unable to do significant damage to Grendel's mother.