I need help on this question.

A sequence of parentheses is balanced if it has equal number of ( and ) symbols and in every prefix of the

sequence the number of left parentheses is greater than or equal to the number of right parentheses. Thus (()) is balanced but

)( is not balanced. The empty sequence is to be balanced.

Consider the grammar S --> (S) | SS | epsilon.

This grammar is claimed to generate balanced parentheses.

1. Prove that any string generated by this grammar is a properly balanced sequence of parentheses.

2. Prove that every sequence of balanced parentheses can be generated by this grammar.