I just want to make sure that I have proved the following exercise correctly.
Given two cardinal numbers $a$ and $b$ where $a$ is infinite. I was to show that $2\le b \le 2^a \implies b^a=2^a$
I will show that $2^a\le b^a\le 2^a$.
Since $2\le b$, we have an injecton $\sigma:\{1,2\}\rightarrow B$.
Letting $H(A,B)$ be defined as the set of all the functions from $A$ to $B$.
Define $\lambda:H(A,\{1,2\})\rightarrow H(A,B), \lambda(f)=\sigma \cdot f$
this is of course an injection and we have $2^a\le B^a$
Similarly we can show that $b^a\le (2^a)^a=2^a$.
Does this look right?
The proof is correct. At the end you might want to say explicitly that you invoke the Cantor-Bernstein theorem to conclude equality.