Gyorgy E. ReveszExercise 1.1: Show the grammar $G$ generates the language $L$ The exercise says 

"Show that the grammar $G = \langle\{S\}, \{a, b\}, S, \{S \to \lambda, S \to aSb\}\rangle$ 
generates the language $L = \{a^i b^i \mid i = 0, 1, 2, \ldots\}$."

Now, I'm new to this subject although I'm familiar with the theory of Turing machines and automata. I'm teaching myself (not in a class, completely independently) from scratch. Please go easy on me.

Now, two questions:

(1) I suspect the way to do this proof is by induction on $i$, which is to say induction on the iterations of letters in the alphabet or the length of the word formed by iterations of $a, b$. Is this assumption correct?

(2) If the assumption in (1) is not correct, what is the correct strategy?
Generally speaking, you need to do two things:


Prove that every word generated by $G$ is in $L$.
Prove that every word in $L$ can be generated by $G$.


Hints for (1): Prove by induction on the length of the derivation.

Hints for (2): Prove by induction on $i$.