Prove that if $a$ is a real number and $a^3 > a$ then $a^5 > a$

Question

Suppose that a is a real number. Prove that if $a^3 > a$ then $a^5 > a$.

Let's consider four cases:

1. $a > 1 $
2. $0 < a < 1$
3. $-1 < a < 0 $
4. $a < -1$

Case 1. $a^3 > a$ and $a^5 > a$

Case 2. $a^3 < a$

Case 3. $a^3 > a$ and $a^5 > a$

Case 4. $a^3 < a$

In all cases where $a^3 > a$, $a^5 > a$ as well. Therefore, $a^3 > a \implies a^5 > a$

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When examining each case, I made assertions without providing any explanation. Because it seems obvious to me, that, for example, if $a>1$ and $n>1$ then $a^n$ will definitely be larger than $a$. However, when writing formal proof, should you make an assumption that it is obvious? Or you need to provide at least some explanation behind the conclusion reached? And in general, is the proof above correct?

Answer 1

This is true, but messier than necessary and lacking justification. Here's an easier way:

Note that $a^2$ is positive, so we can multiply both sides of the inequality $a^3>a$ by it without changing the direction of the $>$, getting that $a^5>a^3>a,$ using the assumption for the last $>$.

On the concept of obviousness: on one hand, sure, it may obvious, given the intuition you likely have from using real numbers for a long time. However, I'd argue that's the point of the exercise (and many simple ones like this).

Answer 2

Solution without cases:

Because $$a^5-a=(a^2+1)(a^3-a).$$

Answer 3

You approach is correct (and two answers posted are also correct, and they use some "shortcuts"). If one is to stay closer to your approach, one may provide some more details, For example, Case 1 could be explained further using that:

If $a>1$ and $b>0$ then $ab>b$. Indeed $ab-b=(a-1)b>0$.

Using the above, with $b=a^3$ and then with $b=a^4$ we get
$aa^3>a^3$ (i.e. $a^4>a^3)$ and $aa^4>a^4$, thus $a^5>a^4>a^3>1$.

There are different ways to provide a valid proof, but in either case you should be able to provide specific details in place of what is obvious. Sometimes it takes some deliberation what would be the best way to prove something, even if it is obvious. That is, to figure what properties of numbers you are formally using.