The meaning of $(\forall M \in \mathbb{R} )( \exists B \in \mathbb{R} )( \forall x>B )( f(x)

Question

I'm trying to understand the meaning of this:

$$(\forall M \in \mathbb{R} )( \exists B \in \mathbb{R} )( \forall x>B )( f(x)<M )$$

the only thing I could figure out that if $x\rightarrow \infty$ then $f(x)$ not going to $+\infty$

Try to set values of $M$ so that you can understand what it going on. For example, set $M = 0$. From the above, we get $B$ such that if $x > B$ then $f(x) < 0$. So $f(x)$ is eventually smaller than $0$ as $x \to \infty$. Now, set $M=-1$, then we get a $B$ such that if $x > B$ then $f(x) < -1$. So $f(x)$ is eventually smaller than $-1$ as $x \to \infty$. Now, change $-1$ to any arbitrary real number, and see if you can say something about $f$ from the given statement. — Sarvesh Ravichandran Iyer, Dec 12 '18 at 07:11
I still dont understand, bucause if I'll take x<B it could be anything, so i can't be bound — shay, Dec 12 '18 at 07:17

score 1 · Answer 1 · edited Dec 12 '18 at 07:23

Graphically, think of $M$ as any horizontal line in the standard coordinate plane. Then your sentence in question is saying that there exists some value $B$ (think of $B$ has a vertical line) such that for all values to the right of $B$, if you evaluate at $f$, they will be below the horizontal line you originally drew ($M$). As the picture below shows, you're essentially saying that 'past $B$, $f$ lives in some quadrant'.

score 0 · Answer 2 · answered Dec 12 '18 at 07:19

This suggests that, after a certain point, $f$ is bounded above by some number, and to any $M$ there is a corresponding "starting point" for this behavior $B$.

Thus, you can choose any upper bound $M \in \mathbb{R}$. Then we have a corresponding bound $B$ on the argument of $f$. For all $x$ beyond that point, then, $f(x)$ is less than this upper bound.

That is, for any upper bound $M$ on $f$, there is a corresponding lower bound $B$ on $x$, such that for any $x > B$, $f(x) < M$.

score 0 · Accepted Answer · answered Dec 12 '18 at 07:49

That's the definition of “$f(x) \to -\infty$ as $x \to \infty$”.

(At least for functions $f \colon \mathbf{R} \to \mathbf{R}$, i.e., functions defined for all $f \in \mathbf{R}$. If the domain is a subset of $\mathbf{R}$, one has to modify the definition slightly.)

The meaning of $(\forall M \in \mathbb{R} )( \exists B \in \mathbb{R} )( \forall x>B )( f(x)

3 Answers3