Question

Show that if the statement P(n) is true for infinitely many positive integers n and$\mathbf{P}\mathbf{(}\mathbf{n}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{\to }\mathbf{P}\mathbf{\left(}\mathbf{n}\mathbf{\right)}$ is true for all positive integers n, thenP(n) is true for all positive integers .

Short answer

The$P\left(n\right)$ is true for all the positive integers n.

Step-by-step solution

Define significance of the contradiction

The contradiction is described as a formula that is always false in nature. Contradiction states that two different or same integers cannot yield a fractional value.

Determination of the truth of the statement

It can be assumed that a positive integern exists that shows that$P\left(n\right)$ does not hold true. Let the smallest integer ism and$P\left(m\right)$ is true for $m>n$. As$P\left(m\right)\to P(m-1)$ holds true and as$P\left(m\right)$ is true, then$P(m-1)$ is also true. According to the definition,$m-1\le n$ . As$m>n$ and the integers differs by the number 1, then $m-1=n$. Then $P\left(n\right)$ holds true that eventually leads to contradiction and also$P\left(n\right)$ is needed to be true for the integersn .

Thus,$P\left(n\right)$ is true for all the positive integersn .