Prove the equivalence of regular expressions

Question

I have a question relating to regular expressions that I'm a bit confused about, If someone can help me out, that would be very much appreciated.

Suppose there exists regular expressions R, S and T.

Then is R* ≡ R* R*?

I know the way to prove this is by showing that L(R*) is a subset of L(R*R*) and vice versa, but I have no idea how to do that.. Just started learning about regular languages so I'm a bit confused. Can I please get any direction on how to do this?

Answer 1

Start with $$ R^*R^* = \left(\bigcup_{i=0}^\infty R^i\right) \left(\bigcup_{j=0}^\infty R^j\right) = \bigcup_{i=0}^\infty \bigcup_{j=0}^\infty R^i R^j = \bigcup_{i=0}^\infty \bigcup_{j=0}^\infty R^{i+j}. $$ As $i,j$ go over all pairs of non-negative integers, $i+j$ goes over all non-negative integers, hence $$ R^*R^* = \bigcup_{i=0}^\infty \bigcup_{j=0}^\infty R^{i+j} = \bigcup_{k=0}^\infty R^k = R^*. $$