Well I woke up in the night, with a terrible fright,
it was the devil at the foot of my bed.
He wore a peach cravate, tied in a Windsor knot,
and it filled my heart with dread.
He said ``Kid, while you snoozed, I guess you missed the news,
of your upcoming change of address.
``You know you've got no use for mathematical truth,
and it's got you in a terrible mess.
``You know your sister does your homework! You're always cutting class bro'!
I've come to take your soul and to hell we're going to go!''
Of course I had to shout, "You've got to give me some out!
I hear the devil's always up for some deal.''
He said with a grin, "They always think they can win!
OK let's play if that's the way that you feel.
``I've got this one little task, you see its all that I ask.
You'll save your soul and plus you'll feel like a hero.
``This sequence I've prepared, it's called one over n squared,
you've got to prove to me its limit is zero.
Now if your proof is air-tight, well then I'll let your soul free,
but if your logic ain't right, kid it belongs to me!''
Well things looked awful grim, you see I'm kinda dim,
I couldn't prove that one plus one equals two.
If only I could glean, what that word `limit' might mean
well maybe then I'd have a clue what to do!
I thought I'd lost the game, but then salvation came
in the form of words ringing out in my head
in my math teacher's voice, which caused me to rejoice
and I told Beelzebub what they said (and I said)
You've got to give me any epsilon. He said ``I'm thinking of an epsilon.''
I've got to come back with a k. He said ``don't keep me all day.''
You choose an n that's more than this. And he said ``follow your bliss.''
I've got to show you that the distance to the limit of the nth term of the sequence is bounded by epsilon!
I broke the silence first: ``OK now do your worst,
give me your epsilon let's get this thing humming.''
He said ``Now slow down, son, that's not the way this thing's done,
it's not my nature to be quite so forthcoming.
``You tell me how your proof would go with it understood
that epsilon is free to follow my choosing.
``But if it does not apply to any value I try
I think you'll find the consequences amusing!
``See Hell is loaded with guys, the only thing that they done
They should have proved it for all, they only proved it for one!''
I said ``How can you expect to get a proof that's correct
with you imposing such an unfair restriction?''
He said ``I don't much care. See I'm not big into fair.
It doesn't show up in my job description.''
Right then and there I knew, no single k would do
for every epsilon with which he'd ensnare me.
If epsilon could range my k would have to change
and as a function of it be free to vary.
I need a formula which on epsilon depends
For the requirements of my diabolical friend.
For me to make a dent in this requirement
I had to know the task for which k was needed.
The only place where k comes into the definition
is the assertion that his n must exceed it.
Though that's a start I suppose, and good as far as it goes,
I'd to hate to be in my shoes at this minute.
So help me out with my mission to keep my soul from perdition,
review with me the definition of limit
Repeat Chorus
To see his villainous grin at the tight fix I was in
had me flummoxed till I started to figure
Sometimes I sneak me a look to the last page of the book
to figure out the guy who pulled the trigger.
To apply this idea to my predicament here
we must repeat the last few lines of the chorus
with general clauses replaced, for our particular case,
but don't get nervous cause I've done this thing for us.
We need from zero to the square of one over n
is less than epsilon, that tricky fellow again.
Nothing makes my blood freeze like inequalities;
I just can't solve them and it drives me to drinking.
But either I've got to try or face eternal hellfire.
Maybe now's the time for me to get thinking.
So by the end of the proof one over n to the twoth
better be less somehow than that Greek letter.
If we apply a square root, and then one over to boot.
The new relation seems to me a bit better
If I could manage to make the number n exceed
one over root epsilon, well then my soul would be freed.
Then Satan pulled back his sleeve, showed off his Rolly to me,
"Look at the time, boy my schedule's a nightmare!
If you've got nothing to say, Then let us be on our way
To your new home, I'd love to show you the sights there!"
My idea gave me some hope, but I was plumb out of rope
I needed time before my moment of testin.
Well here's a hint for you all, when you just need to stall,
Say to your teacher "please repeat the question!"
Repeat Chorus
If you get into a sometime, a situation like mine,
You'll know that panic's not an aid to your thoughts
Satan he runs the show, and he picks n as you know,
So how the hell to make it be what it ought?
I need n to be more than that expression before,
But all he has to make it more than is k.
Wait now something just clicked, k is the thing that I pick!
I'll make it that and then well what will he say?
n would be greater than one over root epsilon,
so by my logic before, I think the proof would be done!
Lucifer drew himself near, and looked at me with a leer,
I knew its now or everlasting bummer.
"Choose epsilon!" I cried, and then his smile died,
"My k's one over the square root of that number"
And then right out of thin air, Archangel Gabriel's there,
He's got his trumpet and he's sporting a halo.
And plus a heavenly choir, those winged chicks was on fire,
they started dancing as Saint G takes a solo.
Everybody dance
I said "choose n as you please, but more than k I believe,
One over n squared's less than epsilon surely.
So it converges to naught, you see I proved what I ought,
Now Mr. Satan why're you looking so poorly?"
He knew my proof was correct, and that it saved me from Heck,
You've never seen a demon looking so grim
He didn't act so cute in his Armani suit
when I said these words to him.
Repeat Chorus