Density Mod1 7264Fb

1. **Problem Statement:** We want to determine if the set $\{f_n \sqrt{2} \bmod 1\}$ is dense in the interval $[0,1]$, where $f_n$ is a sequence of integers. 2. **Key Concept:** A set $\{x_n\}$ is dense in $[0,1]$ if for every $y \in [0,1]$ and every $\epsilon > 0$, there exists some $x_n$ such that $|x_n - y| < \epsilon$. 3. **Important Fact:** If $\alpha$ is an irrational number, then the set $\{n \alpha \bmod 1 : n \in \mathbb{Z}\}$ is dense in $[0,1]$. 4. **Applying to Our Problem:** Since $\sqrt{2}$ is irrational, the set $\{n \sqrt{2} \bmod 1 : n \in \mathbb{Z}\}$ is dense in $[0,1]$. 5. **About $f_n$:** If $f_n$ is a sequence of integers that grows without bound and is not restricted to a finite subset, then the set $\{f_n \sqrt{2} \bmod 1\}$ will also be dense in $[0,1]$. 6. **Conclusion:** Assuming $f_n$ is an integer sequence with infinitely many distinct values, the set $\{f_n \sqrt{2} \bmod 1\}$ is dense in $[0,1]$. **Final answer:** Yes, the set $\{f_n \sqrt{2} \bmod 1\}$ is dense in $[0,1]$ under the assumption that $f_n$ is an integer sequence with infinitely many distinct terms.