Theta asymptotic for $\binom{2m}{m}$

Question

Show that $\binom{2m}{m} = \Theta\left(\frac{2^{2m}}{\sqrt{m}}\right)$ without using Stirling's approximation.

Answer 1

Not a full answer, but some suggestions: the numerator can be rewritten as $$ (2m)!=1 \cdot 2 \cdot \ldots \cdot 2m=1\cdot 3\cdot \ldots \cdot (2m-1)(2 \cdot 4 \cdot \ldots \cdot 2m=(2-1) \cdot (4-1)\cdot \ldots \cdot (2m-1) (2 \cdot 1) \cdot (2 \cdot 2) \cdot (2 \cdot 3) \cdot \ldots \cdot 2m \leq 2^{m} m! \cdot 2^m m!=2^{2m} (m!)^2 $$ Then you remain with $2^{2m}$ as an upper bound. Can you handle from here?