$$\lim_{n \to \infty} \left\{\left(2^{1/2} - 2^{1/4}\right)\left(2^{1/2} - 2^{1/8}\right) \cdots \left(2^{1/2} - 2^{1/(2n+1)}\right)\right\}$$ is equal to

Each term in the product looks like $$(\sqrt{2} - 2^{1/2^{k+1}})$$.

As $$k$$ increases, $$2^{1/2^{k+1}}$$ gets closer and closer to $$2^0$$, which is 1.

So, every term in the sequence eventually settles at approximately $$\sqrt{2} - 1 \approx \mathbf{0.414}$$.

When you multiply a number less than 1 (like 0.414) by itself infinitely many times, the result always shrinks to 0.

Mathematically, if $$|a| < 1$$, then $$\lim_{n\to\infty} a^n = 0$$. Since all terms here are positive and less than 1, the product must converge to zero.