I'm supposed to proove the proposition that for all integers k>=2, that k^2<k^3. The book says you can use induction or another method. I can't seem to get either fully proved. Any help would be great. Thanks!!
This is actually easier without induction
Basically, $\displaystyle 1<n$ since $\displaystyle n>2$ and we "know" that for $\displaystyle p>0$, if $\displaystyle m<n$ then $\displaystyle pm<pn$
Also, $\displaystyle n^2>0$ since $\displaystyle n>2$
So letting $\displaystyle n^2=p$ and $\displaystyle 1<n$, then $\displaystyle n^2\cdot 1<n\cdot n^2$
Now of course, you might not acutally "know" the proposition that I referenced if you haven't proven it yet
You may also do it this way:
$\displaystyle k^2<k^3$ is equivalent to $\displaystyle \frac{k^2}{k^3} < 1 \Rightarrow \frac{1}{k}<1$ (we can do that since $\displaystyle k\neq 0$)
Now, $\displaystyle \frac{1}{k}<1$ is obviously true for any $\displaystyle k>1$...