TIFR ECE 2017 | Question: 5

Question

Consider the inequality
\[
n-\frac{1}{n} \geq \sqrt{n^{2}-1},
\]
where $n$ is an integer $\geq 1$. Which of the following statements is $\text{TRUE?}$

• A. This inequality holds for all integers $n \geq 1$
• B. This inequality holds for all but finitely many integers $n \geq 1$
• C. This inequality holds for only finitely many integers $n \geq 1$
• D. This inequality does not hold for any integer $n \geq 1$
• E. $n-\frac{1}{n}=\sqrt{n^{2}-1}$ for infinitely many integers $n \geq 1$