Proving sequences that tend to infinity - Real Analysis

Question

If $b_n\to\infty$ and $\{a_n\}$ is such that $b_n>a_n$ for all $n$, then $a_n\to\infty$.

We are to prove this by using either a)formal definitions or b)counter examples.

I am very unsure of how to prove using the formal definitions. I can see it is quite clear that if I chose to use a counter example to prove we can't show that $a_n$ goes to infinity simply because it is less than $b_n$, that this would maybe be easier, but once again, I am at a loss on how to go about setting up this proof.

Any help in proving this would be great.

Answer 1

I assume you meant $a_n \geq b_n$. A sequence $(x_n)_{n \in \mathbb{N}}$ diverges to $\infty$ if for every $K$ there exists an $N \in \mathbb{N}$ such that for all $n\geq N$ it holds that $x_n \geq K$. Now, we already know that $(b_n)_{n \in \mathbb{N}}$ satisfies this property. Now given an arbitrary $K$ how do we choose $N$ such that $a_n \geq K$ all $n \geq N$ using that $a_n \geq b_n$?

Answer 2

Your statement is false. as a counter example,

take

$$b_n=n \text{ and } \; a_n=-n.$$

then $$a_n\le b_n,$$

$$b_n\to +\infty$$ but $$a_n\to -\infty$$