What Methods Can Be Used to Prove Sequence Divergence?

[Bipolarity]

I'm trying to understand divergence of a sequence (not series). What methods can I use to prove divergence? I know that convergence can be proven using various methods, such as squeeze theorem and sum, difference, product and quotient rule etc.

Could I use the following to prove divergence?

If $a_{n}$ is a sequence of real numbers, $f(n) = a_{n}$ and $\lim_{n→∞} f(n)$ does not exist, but is not equal to ∞ or -∞, does $a_{n}$ necessarily diverge?

If $a_{n}$ is a sequence of real numbers, $f(n) = a_{n}$ and $\lim_{n→∞} f(n) = ∞$, does $a_{n}$ necessarily diverge?

These two ideas will greatly facilitate my understanding of sequence divergence.
Thanks!

BiP

 

[micromass]

Yes to both questions.

 

[Bipolarity]

Yes to both questions.

Hey micro, but what about the sequence $a_{n} = sin(2πn)$. It is the case that
$\lim_{n→∞}f(n)$ does not exist, yet the limit of $a_{n}$ converges to 0, right??

BiP

 

[micromass]

The limit $\lim_{n\rightarrow +\infty} f(n)$ does exist and is zero. (I assume that n is always an integer)

However, if you extend f to $f(x)=\sin(2\pi x)$ for $x\in\mathbb{R}$, then the limit $\lim_{x\rightarrow +\infty} f(x)$ doesn't exist.

 

[pwsnafu]

$\lim_{n→∞}f(n)$ does not exist

Why do you say that?

Edit: ninjaed

 

[Bipolarity]

The limit $\lim_{n\rightarrow +\infty} f(n)$ does exist and is zero. (I assume that n is always an integer)

However, if you extend f to $f(x)=\sin(2\pi x)$ for $x\in\mathbb{R}$, then the limit $\lim_{x\rightarrow +\infty} f(x)$ doesn't exist.

micromass, I'm sorry I think I misphrased my question. When I refer to f(n) in my original post, I refer to it as a function with domain ℝ as opposed to $a_{n}$ which I take to be defined only for natural numbers.

Given this clarification, which of the following original statements is true and why?

BiP