Overleaf

\textbf{Claim:} The set of positive integers that contains the integer 1 and has the property that if it contains all the positive integers 1, 2, ..., k, then it also contains the integer k+1, must be the set of all positive integers.

\textbf{Proof by Principle of Mathematical Induction (PMI):}

\textbf{Step 1: Base Case:} Let k = 1. In this case, the set contains only the integer 1, and it trivially contains the integer 2 (k+1 = 1+1 = 2). Therefore, the property holds for the base case.

\textbf{Step 2: Inductive Hypothesis:} Assume that the property holds for some positive integer n, where n ≥ 1. That is, if the set contains all the positive integers 1, 2, ..., n, then it also contains the integer n+1.

\textbf{Step 3: Inductive Step:} We need to prove that the property holds for n+1. That is, if the set contains all the positive integers 1, 2, ..., n+1, then it also contains the integer (n+1)+1 = n+2.

By the inductive hypothesis, we know that the set containing all the positive integers 1, 2, ..., n also contains the integer n+1. Now, if we add the integer n+2 to this set, it will still contain all the positive integers 1, 2, ..., n+1. Therefore, the property holds for n+1.

By the principle of mathematical induction, we have shown that if the set contains the integer 1 and if it contains all the positive integers 1, 2, ..., k, then it also contains the integer k+1, for all positive integers k. Therefore, the set of positive integers satisfying this property is the set of all positive integers.

\qed

History

Overleaf was selected as one of the ten teams who participated to the 2013 Summer's Bethnal Green Ventures (BGV) accelerator programme. That program started on the July 1, 2013, and lasted for 3 months. The Demo Day of that BGV 2013 Summer program was held on the September 19, 2013.

The company received strategic investment from Digital Science in 2014. Overleaf won Innovative Internet Business at the 2014 Nominet Internet Awards, and featured 99th in SyndicateRoom's 2018 list of Britain's top 100 fastest-growing business.

Overleaf has been discussed as a tool for writing scientific publications in Nature, Science, Red Hat's opensource.com and the German IT magazine Heise Online. "In 2017, CERN, Europe’s particle-physics laboratory near Geneva, Switzerland, adopted Overleaf as its preferred collaborative authoring platform."

Overleaf provides templates for submission to scientific journals and conferences. For example, the IEEE and Springer (incl. Nature) mention the possibility for submission using Overleaf.

\textbf{Claim:} The set of positive integers that contains the integer 1 and has the property that if it contains all the positive integers 1, 2, ..., k, then it also contains the integer k+1, must be the set of all positive integers.

\textbf{Proof by Principle of Mathematical Induction (PMI):}

\textbf{Step 1: Base Case:} Let k = 1. In this case, the set contains only the integer 1, and it trivially contains the integer 2 (k+1 = 1+1 = 2). Therefore, the property holds for the base case.

\textbf{Step 2: Inductive Hypothesis:} Assume that the property holds for some positive integer n, where n ≥ 1. That is, if the set contains all the positive integers 1, 2, ..., n, then it also contains the integer n+1.

\textbf{Step 3: Inductive Step:} We need to prove that the property holds for n+1. That is, if the set contains all the positive integers 1, 2, ..., n+1, then it also contains the integer (n+1)+1 = n+2.

By the inductive hypothesis, we know that the set containing all the positive integers 1, 2, ..., n also contains the integer n+1. Now, if we add the integer n+2 to this set, it will still contain all the positive integers 1, 2, ..., n+1. Therefore, the property holds for n+1.

By the principle of mathematical induction, we have shown that if the set contains the integer 1 and if it contains all the positive integers 1, 2, ..., k, then it also contains the integer k+1, for all positive integers k. Therefore, the set of positive integers satisfying this property is the set of all positive integers.

\qed

See also

• Collaborative real-time editor
• Comparison of TeX editors
• TeX – Typesetting system