Is there an opposite to the Squeeze theorem?

Question

I'm familiar with the squeeze theorem (AKA Two Policemen and a Drunk---no matter how wobbly, the drunk will reach the same destination as the policemen). Is there is an opposite theorem that tugs or pulls (AKA One Policeman hand-cuffed to a Drunk---no matter how wobbly, the drunk will also reach the same destination)? Squeeze, Pinch, etc.

A better analogy than the hand-cuffs scenario: You start a walk with your dog on an extendible leash at full extension. At each step, you shorten the leash a couple of inches. When you reach your destination your dog is tightly controlled.

Could you perhaps be a little more specific about which theorem you're referring to -- perhaps even state it? There are several places in mathematics (usually in branches of analysis) that have theorems that could be called "the squeeze theorem". For extra points, you should also attempt to state the "one policeman hand-cuffed to a drunk" theorem you're proposing. — hmakholm left over Monica, May 09 '15 at 14:05
Hmm, I still see no statement of a theorem -- only an analogy and no indication of what mathematical content you consider it to be an analogy for. — hmakholm left over Monica, May 09 '15 at 14:24
@HenningMakholm; is it if $a_n\le c_n\le b_n, ;\forall n$, and $a_n\to k, b_n\to k$, then $c_n\to k$? — JMP, May 09 '15 at 14:28
@JonMarkPerry: It might be, but it might be something different. I don't know -- that's why I'm asking the OP. — hmakholm left over Monica, May 09 '15 at 14:29

score 12 · Answer 1 · answered May 09 '15 at 14:04

12

Here is a possible candidate, which can be interpreted as a push theorem (probably not a pull theorem).

If $a_n \leq b_n$ and $a_n \to \infty$, then $b_n \to \infty$.

Similarly, if $a_n \leq b_n$ and $b_n \to -\infty$, then $a_n \to -\infty$.

answered May 09 '15 at 14:04

Adhvaitha

20,259

score 5 · Accepted Answer · answered May 09 '15 at 15:20

Your two examples merely seem like variations of the squeeze theorem.

Cop pulls drunk example:

If $|a_n - b_n| \leq M$ for all $n \geq N$, and $a_n \to \infty$ then $b_n \to \infty$.

Master pulls dog example:

If $|a_n - b_n| \to 0$ and $a_n \to M$ then $b_n \to M$.

Is there an opposite to the Squeeze theorem?

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