- #1

cragar

- 2,552

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## Homework Statement

Prove that there is no function f:N→N such that

for all [itex] n \in N [/itex] , f(n)>f(n+1).

There is no infinite decreasing sequence of naturals.

## The Attempt at a Solution

Let's assume that their is a function f(n)>f(n+1) for all n.

This would also imply that f(n) will be mapped to all of N using all n in N.

There exists some x in N such that f(x) equals the first natural number.

if f(x) gets mapped to the first natural then f(x+1) gets mapped to some natural.

This is either the first natural or some larger natural. But this is a contradiction

because we assumed that f(n)>f(n+1). so therefore no function exists.