Question about adding spells to your spellbook when multiclassing.

I'm currently running a champaign in 5e and one of my players is a lvl 2 sorcerer / lvl 1 wizard character who recently found a book of spells in our latest adventure. One of the spells in the book is a level 2 spell that he's asked if he can learn. I was initially skeptical since he's only a lvl 1 wizard, but when I read the rules described in the players handbook I wasn't so sure (hence my asking).

From my read in the players handbook it sounds like a wizard can add a spell to his spellbook as long as he has spell slots of that spells level or higher. Now, since 5e multi-classing combines the spellslots of both sorcerers and wizards his character has level 2 spell slots, does this mean he can learn the spell?