query: Show that if $a \in R$ has more than one left-inverse then it has infinite.
positive_passages.docid: 21243
positive_passages.text: Kaplansky's theorem of infinitely many right inverses in monoids?
negative_passages.docid: 136321
negative_passages.text: Show that if $\sup\big\{\sum\lvert\, f(a)\rvert\big\}  0\}$ is countable.
negative_passages.docid: 38242
negative_passages.text: Explicit bijection between the successor of $\alpha$ and $\alpha$ (infinite ordinal).
negative_passages.docid: 10702
negative_passages.text: Show that $\left\{A\subset\Omega: A~\text{ is countable or }A^C\text{ is countable}\right\}$ is a $\sigma$-algebra
negative_passages.docid: 155579
negative_passages.text: Show that $End_A(A)$ = {$r_a$ | $a ∈ A$}
negative_passages.docid: 50019
negative_passages.text: Prove $0$ is a partial limit of $a_n$
negative_passages.docid: 33030
negative_passages.text: Proof that $\inf A = -\sup(-A)$
negative_passages.docid: 145428
negative_passages.text: Set $A$ is countably infinite if and only if there exists a bijection $f: \mathbb{N} \rightarrow A$
negative_passages.docid: 154240
negative_passages.text: Inverse of Heine–Cantor theorem
negative_passages.docid: 66951
negative_passages.text: Proving the existence of a sequence of functions converging to 0 with an added property
negative_passages.docid: 152784
negative_passages.text: $f \in {\mathscr R[a,b]} \implies f $ has infinitely many points of continuity.
negative_passages.docid: 9021
negative_passages.text: An infinite subset of a countable set is countable
negative_passages.docid: 107641
negative_passages.text: Show that an infinite set of families are linearly independent
negative_passages.docid: 59133
negative_passages.text: Show that an inverse of a bijective linear map is a linear map.
negative_passages.docid: 78763
negative_passages.text: Prove that $ S=\{0\}\cup\left(\bigcup_{n=0}^{\infty} \{\frac{1}{n}\}\right)$ is a compact set in $\mathbb{R}$.
negative_passages.docid: 132868
negative_passages.text: Bijection between an infinite set and its union of a countably infinite set
negative_passages.docid: 99633
negative_passages.text: Let $α$ be an ordinal and $A$ be a set of ordinals. Then $\sup\limits_{β∈A} (α+β) = α+\sup\limits_{β∈A}(β)$