Is there an opposite to the Squeeze theorem? 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.

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 $b_n \to M$.