query: For an assertion $P(n)$ where $n\ge2$, the following information is given,
positive_passages.docid: 12857
positive_passages.text: Induction proof: $S$ contains powers of 2 and predecessors implies $S={\bf N}$
negative_passages.docid: 84930
negative_passages.text: How to justify $\phi(n) \ge \sqrt{n}$
negative_passages.docid: 95000
negative_passages.text: Using induction to prove that $2 \mid (n^2 − n)$ for $n\geq 1$
negative_passages.docid: 151935
negative_passages.text: Proving the predicate $\forall n \in \mathbb{N} , n \geq 2$ using the Pigeonhole principle
negative_passages.docid: 157898
negative_passages.text: Showing that $\sum_{i=1}^n \frac{1}{i} \geq \log{n}$
negative_passages.docid: 130093
negative_passages.text: Prove by Induction : $n < 2^n$
negative_passages.docid: 62996
negative_passages.text: A problem regarding the proof of ${p^nk\choose p^n}\equiv k\mod p$, where $p\nmid k$.
negative_passages.docid: 20964
negative_passages.text: Proof that $n^2 < 2^n$
negative_passages.docid: 21250
negative_passages.text: Prove $x \equiv a \pmod{p}$ and $x \equiv a \pmod{q}$ then $x \equiv a\pmod{pq}$
negative_passages.docid: 103732
negative_passages.text: Proving that for $n \equiv 0 \pmod{2}$, we get ${n \choose 0} < {n \choose 1} <\ldots< {n \choose n/2-1}<{n\choose n/2}$ etc.
negative_passages.docid: 149758
negative_passages.text: How to prove $\log n < n$?
negative_passages.docid: 92569
negative_passages.text: Showing that ${\rm E}[X]=\sum_{k=0}^\infty P(X>k)$ for a discrete random variable
negative_passages.docid: 91197
negative_passages.text: Proving $n! \ge 2^{n-1 }$for all $n\ge1 $by mathematical Induction
negative_passages.docid: 4970
negative_passages.text: For $n \geq 2$, show that $n \nmid 2^{n}-1$
negative_passages.docid: 52614
negative_passages.text: (exercise from Tao's analysis book) Proof of a lemma relating to power set of X
negative_passages.docid: 52157
negative_passages.text: Prove that $d(n)\leq 2\sqrt{n}$
negative_passages.docid: 76347
negative_passages.text: Shorter proof for some equvalences