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How I can prove that this: $$ \frac{1}{2} < \frac{2}{3} < \frac{3}{4} < \frac{4}{5} < \frac{5}{6}<...<\frac{n+1}{m+1}<...$$ always holds when $n < m$ and $n$ and $m$ are positive integers?

I started writing this $\frac{n}{m}<\frac{n+1}{m+1}$ but then I don't know how to proceed.

Rafa_Asp1
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2 Answers2

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Because $$\frac{n+1}{m+1}-\frac{n}{m}=\frac{m-n}{m(m+1)}>0$$ for $m>n>0.$

Michael Rozenberg
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This a high-school standard exercise on inequalities:

As all numbers are positive, $$ \frac nm<\frac{n+1}{m+1}\iff n(\not m+1)<m(\not n+1)\iff n<m. $$

Bernard
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    You answered an identical question before: https://math.stackexchange.com/a/3373570/42969. – Martin R Oct 03 '20 at 09:05
  • @MartinR: I didn't remember. Anyway, I generalised the answer in this other post. Here I tried to make it as short and expressive as possible. – Bernard Oct 03 '20 at 09:39