This is a useul theorem of Analysis, to prove a number x is not greater than ano

Question

This is a useul theorem of Analysis, to prove a number x is not greater than another number y. Please read the following indented proof and try to translate it to a linear proof in plain English. Then try to write the statement of the theorem that is proven, (due to typesetting issues, the proof is lacking the lines of arguments: there are three of them, draw the lines of argument before answering the question.) Also please note that here is a proof structure, known as contraposition: (it says if we assume q and reach p then we can conclude that p Rightarrow q) q p p Rightarrow q now the proof: Given real numbers x and y (x lessthanorequalto y) (x lessthanorequalto y) y

Explanation / Answer

The statement for the theorem proved here is -

for every number n belonging to set of natural numbers , there exists another number m such that n <m .

This theorem is also called Archimedian principle whose exact statement in analysis is given as

For any €>0 , there exists m belonging to set of natural numbers such that 1/n < € for all n>= m

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