Repeating Decimal Frustration

I've always been frustrated explaining, then trying to convince students that

That is, a decimal point followed by an infinite number of nines, is exactly equal to 1.  The contrary students usually offer an argument that you can never write enough nines to "get to" 1.  Usually, when confronted with such a stubborn student, I would rely on my years of teaching experience, and my well developed sense of patience and I would yell; "SHUT UP...IT IS TOO EQUAL".  The stubborn student would predictably reply; "Dude...WhatEver".  To further help explain the concept, I would grit my teeth together until my face turned bright red, simultaneously stomp both feet and yell, just a little louder than last time; "SHUT UP...IT IS TOO EQUAL".  

I've found (again, relying on my years of teaching experience) that screaming louder and louder was successful at raising my blood pressure considerably, but only the best "top notch" students were able to absorb and understand this fact solely because of my increased volume.  

Here's a short list of some explanations and reasons why point nine repeating equals one for those who don't learn well by the "volume method".

1.)  If you can swallow the fact that :

then just multiply both sides by 3 and you have your result.  This explanation works fine, but is a little unsatisfying for some reason.

2.) Try this:  

Let x = .9999....

then 10x = 9.9999...

Subtracting these two equations gives:               9x = 9, 

so x=1.

Notice x = .9999.... and also x = 1, therefore .9999.... = 1.

3.)  For any two different real numbers, there are an infinite number of other real numbers in between.  (search for Archimedes property to see more formal and detailed info)  If you search between .99999.... and 1, you won't be able to find any numbers.  Therefore .9999... and 1 are not distinct.  They are equal.

4.)  Calculate the geometric series .9 + .09 + .009 + .0009 + .00009 + .....  This series can be represented with the notation: 

Recall the formula for calculating a geometric series:

So in our case:

(Search for geometric series for more detailed info)

In conclusion: Choose your favorite explanation, or:  "SHUT UP... IT IS TOO EQUAL!!"