I am rather new to mathematical induction. Specially inequalities, as seen here How to use mathematical induction with inequalities?. Thanks to that question, I've been able to solve some of the form $ 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \leq \frac{n}{2} + 1 $.
Now, I was presented this, for $n \ge 4$:
$2^n
I tried to do it with similar logic as the one suggested there. This is what I did:
Prove it for $n = 4$: $2^4 = 16$ $4! = 1\cdot2\cdot3\cdot4 = 24$ $16 < 24$ Assume the following: $2^n
- So first we take $2^{n+1}$ which is equivalent to $2^n\cdot2$
- By our assumption, we know that $2^n\cdot2 < n!\cdot2$
- This is because I just multiplied by $2$ on both sides.
- Then we'll be finished if we can show that $n! \cdot 2 < (n+1)!$
- Which is equivalent to saying $n!\cdot2
- Since both sides have $n!$, I can cancel them out
- Now I have $2<(n+1)!$
- This is clearly true, since $n \ge 4$
Even though the procedure seems to be right, I wonder:
- In the last step, was it ok to conclude with $2<(n+1)!$? Was there not anything else I could have done to make the proof more "careful"?
- Is this whole procedure valid at all? I ask because, well, I don't really know if it would be accepted in a test.
- Are there any points I could improve? Anything I could have missed? This is kind of the first time I try to do these.