$Q(n)$: "$P(k)$ holds for all $k<n$". Then why is $Q(0)$ clearly true?
126713
Strong induction and vacuous truth
31930
Given the sequence $(x_n)$ is unbounded, show that there exist a subsequence $(x_{n_k})$ such that $\lim(1/x_n)=0$.
3599
show that if ${a_n} \to a$, and $a>0$, then $\exists N$ such that $a_n>0$ for $n \ge N$.
137511
Trying to show that $\sup \{b^p:p\in Q,\;0<p<x\} = \inf\{b^q:q\in Q,\;x<q\}$
83407
How to show that that $q \mid p-1$?
103314
Let $\{K_n \}_{n \in \mathbb{N}}$ be a collection of sets, where $K_n = [n, \infty)$. Why is $\bigcap_{n \in \mathbb{N}} K_{n}$ empty?
95860
The limit of the difference of two consecutive sequence members is equal to $0$. Can we conclude that the sequence itself has a limit?
113342
Confusion of the decidability of $(N,s)$
8360
$\sum_i x_i^n = 0$ for all $n$ implies $x_i = 0$
151935
Proving the predicate $\forall n \in \mathbb{N} , n \geq 2$ using the Pigeonhole principle
58663
How to show that if $\phi(n)$ equals to n itself, then n must be 1?
118674
Simple upper bound for $\binom{n}{k}$
59187
Proof, is $\forall x : (P(x) \lor Q(x)) \Leftrightarrow \forall x : P(x) \lor \forall x : Q(x)$?
105110
Why does $\sqrt{n\sqrt{n\sqrt{n \ldots}}} = n$?
14688
Do sequences in $\ell^p$ belong to some $\ell^q$ for $q<p$?
3786
Help finding a combinatorial proof of $k {n \choose k } = n {n - 1 \choose k -1}\;$
79187
Can we find $c_n$ to make $X_nc_n \rightarrow 0$ almost surely, even if $EX_n > \infty$?