I am asking if we can deduce the following inequality: $∏_{k=1}^{∞}4k²/(4k²-1)×u_{k}<∏_{k=1}^{∞}4k²/(4k²-1)×a$

Question

We know that $$∏_{k=1}^{∞}4k²/(4k²-1)=π/2$$

Assuming we have an inequality of the form $$0<u_{k}<a<1$$

I am asking if we can deduce the following inequality:

$$∏_{k=1}^{∞}4k²/(4k²-1)×u_{k}<∏_{k=1}^{∞}4k²/(4k²-1)×a$$

Answer 1

It's clearly false, because $$ \prod_{n = 1}^{\infty} \frac{4 n^2}{4 n^2 - 1} a = \lim_{n \to \infty} \left(\left(\prod_{k = 1}^n \frac{4 k^2}{4 k^2 - 1}\right) \left(\prod_{k = 1}^n a\right)\right) = \lim_{n \to \infty} \left(\prod_{k = 1}^n \frac{4 k^2}{4 k^2 - 1}\right) \lim_{n \to \infty} a^n = \frac{\pi}{2} \cdot 0 = 0\mbox{.} $$