The sorcerer's spells known feature says:
You know two 1st-level spells of your choice from the sorcerer spell list.
The Spells Known column of Table: The Sorcerer shows when you learn more sorcerer spells of your choice. Each of these spells must be of a level for which you have spell slots. For instance, when you reach 3rd level in this class, you can learn one new spell of 1st or 2nd level.
The part I'm concerned with is the following: "Each of these spells must be of a level for which you have spell slots." The wording, unfortunately, assumes single-class characters only.
We know from the multiclass rules that if a character is multiclassed between two casting classes, that spell slots are granted based on the following:
Spell Slots: You determine your available spell slots by adding together all your levels in the bard, cleric, druid, sorcerer, and wizard classes, and half your levels (rounded down) in the paladin and ranger classes. Use this total to determine your spell slots by consulting Table: Multiclass Spellcaster.
A level 5 wizard who takes 1 level in Sorcerer has four Level 1 spell slots, three Level 2 slots, and three Level 3 slots, per the multiclass spellcaster table (visible here).
If I'm reading the sorcerer class entry correctly, the first level of sorcerer requires that you pick two Level 1 spells ("You know two 1st-level spells of your choice from the sorcerer spell list."), but every sorcerer level taken after the first can allow the character to learn spells of any level he has slots in, including Level 3 spells such as Fireball.
Is my assumption correct? Can I multiclass into Sorcerer from Wizard and, aside from the initial Level 1 spells known requirement (which can later be swapped with higher level spells per the sorcerer class feature), add spells to my list of Sorcerer Spells Known of any level I have slots in from the multiclass spell slots table?