math analysis - sequence proof?

Question

I'm struggling getting started with a problem:

for all real number x, [x] is defined as the greatest integer <=x;
i.e. the unique integer such that x-[x] is an element of [0,1)
for example [3.4] = 3 and [-3.4] = -4

let a be an irrational number, and define a real sequence {x_n} by setting

x_n = na-[na]

Prove:
the values of this sequence are distinct irrational numbers in the interval (0,1)

that's where I'm stuck at, I think it has to do with either showing x_n is monotone decreasing, or using the that x-[x] is in [0,1)

Answer 1

You need to show the following:

1. You know that y - [y] is always in [0,1), so you need to show that x_n can never be 0.

2. Can na ever be rational?

3. For distinct positive integers m and n, assume na - [na] = ma - [ma]. You should be able to get a contradiction.

Answer 2

until now you're equipped to settle for an data in arithmetic, you may desire to have a sparkling tips of what the words in touch mean. What I want you to understanding on is, what *is* .9999...? Or any decimal growth, like 3.1415... or something. A volume does not distinction. that is now not 0.9 one 2d and then 0.ninety 9 right here, as though some exterior guy or woman is changing it via employing taking into attention it. that is caught someplace on the quantity line, inspite of the type you describe it. And if I ask, what's the replace between a million and 0.9999...., it is going to have have been given to be a relentless volume too, now not some "infinitely small" element. There must be an respond; you will generally subtract numbers. And it somewhat is easy to look that the replace, if advantageous, might have have been given to be smaller than another advantageous volume. (its decimal growth might desire to start with extra zeros 0.0000...) If now not advantageous, it would desire to be 0. Can that is that there is a advantageous volume smaller than another advantageous volume? No, then it would desire to be smaller than itself (or area itself). Contradiction. right that's a distinctive viewpoint: do you sense that a million/2 + a million/4 + a million/8 + a million/sixteen + ...=a million? (classic Achilles & the Hare downside.) it is comparable to announcing 0.11111.... in binary is same to a million. Your downside is an identical suggestion base ten.

Answer 3

This sounds like a "proof by contradiction": Assume that two numbers in the sequence are equal, say x_j and x_k. Then show that this means that a must be some rational number ...