For $n \in\mathbb N$
I have to prove, using mathematical induction: $\forall n\in\mathbb N(n<2^n)$
It holds for $n=1$
So I assume $\forall n\in\mathbb N(n<2^n)$ alright.
I need to prove the following then: $\forall n\in\mathbb N(n+1<2^{n+1})$
This is how I prove it:
By our assumption, we know that $n+1 < 2^n+1$ (Just added 1 to both sides)
So it is now enough to prove that
$2^n+1 < 2^{n+1}$
Which is equivalent to proving the following:
$2^n+1<2^n\cdot2$
Alright. If $n\ge1$ then it might seem obvious that the above holds. I have some questions, however:
- Isn't there a way to make it more "obvious"?
- When "updating" $n$ to $n+1$, I don't have to do it for the $\forall n$ part right?
- You know when I added a $1$ to both sides? Honestly I don't even know why/how did I do that - it just worked at the end. Can you explain to me a better way to come up with the proper "modifications"?