How would you go about proving the following statement

$\displaystyle \forall x \in \mathbb{N} ((\forall k : 2 \leq k \leq x) \rightarrow x+k \leq kx) $

Would you use strong induction ?? Could one explain how ?

thank you

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- Feb 23rd 2013, 10:34 AMbaxy77baxProving forall statement
How would you go about proving the following statement

$\displaystyle \forall x \in \mathbb{N} ((\forall k : 2 \leq k \leq x) \rightarrow x+k \leq kx) $

Would you use strong induction ?? Could one explain how ?

thank you - Feb 23rd 2013, 10:49 AMOptikalRe: Proving forall statement
I don't think that induction is needed for this problem. I would prove it as follows:

Suppose that x is in N, and that k is a number satisfying 2<=k<=x.

Then, x+k<=x+x=2x<=kx as required.

The first inequatlity follows from the fact that k<=x and the last inequality follows from the fact that 2<=k.

Hope this helps!

Optikal