You can see how the nameof the theoremmakessensefromthe statement.Aftera certainpointin the
sequences,the termsof a sequencecnare sandwichedor squeezedbetweenthe termsof two convergent
sequenceswith the samelimit. Thenthe limit of the sequencecnis squeezedto becomethe sameas the
limit of the two convergentsequences.Let’s look at an example.
Example 13
Prove.
Solution
Recallthatn! is read as “n factorial”and is writtenasn! =n× (n- 1) × (n- 2)× ... ×1.
We wantto applythe Sandwichtheoremby squeezingthe sequence betweentwo sequencesthat
convergeto the samelimit.
First,we knowthat. Now we wantto find a sequencewhosetermsgreaterthan or equalto the
termsof the sequence for somen.
We can write
Sinceeachfactorin the product is less thanor equalto 1, thenthe product
. Thenwe makean inequality:
Thus,. By usingthe RulesTheorem,we have
and. Thus,. By the