query: Prove $n! \geq 2^{n - 1}$ ; $n\geq 0$
positive_passages.docid: 16798
positive_passages.text: Prove the inequality $n! \geq 2^n$ by induction
negative_passages.docid: 88055
negative_passages.text: Use induction to prove the following: $1! + 2! + .... + n! \le (n + 1)!$
negative_passages.docid: 24469
negative_passages.text: Prove $ 2<(1+\frac{1}{n})^{n}$
negative_passages.docid: 52157
negative_passages.text: Prove that $d(n)\leq 2\sqrt{n}$
negative_passages.docid: 126889
negative_passages.text: Proof by induction: inequality $n! > n^3$ for $n > 5$
negative_passages.docid: 145519
negative_passages.text: Prove that for any natural number $n>1$, we have $(2n)!<(n(n+1))^n$
negative_passages.docid: 116387
negative_passages.text: How to prove that $\Gamma(x) \leq x^x$ for $x \geq 1$
negative_passages.docid: 149056
negative_passages.text: Prove that for all $n\geqslant 1$ we have $F_n<{\left(\frac 74\right)}^n$.
negative_passages.docid: 138819
negative_passages.text: Prove that $\phi(n) \geq \sqrt{n}/2$
negative_passages.docid: 39073
negative_passages.text: Proving that: $\forall n \in \mathbb{N} :|f^{(n)}(x)|\leq \frac{1}{n+1}$
negative_passages.docid: 141837
negative_passages.text: Proof of $\sup\{ (n!)^{(\frac{1}{n})}| n ∈ N\} = +∞?$
negative_passages.docid: 10902
negative_passages.text: Showing that $ \prod\limits_{i=0}^{n} {n \choose i}\leq(\frac{2^n-2}{n-1})^{n-1} $
negative_passages.docid: 45914
negative_passages.text: Proving that $n!≤((n+1)/2)^n$ by induction
negative_passages.docid: 73987
negative_passages.text: Prove the inequality $({1+\frac{a}b})^n$ + $(1+\frac{b}a)^n$ $\geq$ $2^{n+1}$
negative_passages.docid: 145891
negative_passages.text: Prove that $n! > \sqrt{n^n}, n \geq 3$
negative_passages.docid: 136274
negative_passages.text: $n! \leq \left( \frac{n+1}{2} \right)^n$ via induction
negative_passages.docid: 21612
negative_passages.text: How to prove this form of $n$?