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Proof by Deduction

Proof by Deduction

This method of proof is essentially directly showing the truth of a statement.

That is a chain of logic something of the form:

"if a is true then b is true".

"If b is true, then c is true."

"since a is true, then c must be true".

 

Proof by Deduction 1

Given that $k(n) = n^2 + 4n + 7$ Prove that for all $n\in\mathbb{Z}$, $k(n) \gt 0$
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The function $k(n)$ can be rewritten as $$k(n) = n^2+ 4n + 7 = (n+2)^2 + 3$$

For all $n$, $(n+2)^2 \geq 0$ as squaring an integer gives either a positive answer or zero.

Therefore $(n+2)^2 + 3 \gt 0$ and hence $k(n)\gt 0$ for all $n$