What is the meaning of $n\in \aleph$

Question

Using mathematical induction, prove that, for each $n\in \aleph$

$$n<3^n$$

Dear all, what is the meaning of "$n\in \aleph$" . How to substitute it and prove that? please give me one step or some basic information about that. what to do?

Answer 1

Presumably it means $n \in \mathbb{N}$.

$\aleph_0$ is the first infinite cardinal, it's the cardinality of the natural numbers. In the most common formalism of set theory, it's identified with $\omega$, the first infinite ordinal number, which in turn is identified with $\mathbb{N}$. So when you see $n \in \aleph_0$ what it means is $n \in \mathbb{N}$. I'm guessing that here $\aleph$ is written instead of $\aleph_0$. This is in line with the meaning of 'mathematical induction' (as opposed to other induction principles).

It seems to be an odd use of the notation, though.

In fact $\kappa < 3^{\kappa}$ is true for all cardinals $\kappa$, so it doesn't really matter what the author means by $\aleph$ if you want to prove this more general result.

Answer 2

$n<3^{n}$
step 1. for $n=1$
$n<3^{n}$
step 2. $1<3^{1}$
$1<3$
It is the proof

Thanks for all.....Just I prove that. Please tell me I am correct or not? what are the missing steps? Sorry for my bad English....