is Set of Real Numbers Between 0 and 1 really uncountably Infinite? [closed]

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Cantor's Set of Countable infinite and Uncountable infinite Infinites

You may know and you may have proved that Set of Real Numbers Between 0 and 1 are Uncountably Infinite. Mean we Can not Map Every number of that set on a different Natural Number.

I got a Technique by which I would be able to Map all Real numbers between 0 and 1 on a different Natural Number. Technique is Simple Replace the Decimal Point with 1 and Map the Original on that Number Such that Map 0.0003 on 10003 and 0.03 on 103

By using this Technique we Would be able to Map all Real Numbers Between 0 and 1 on Natural Numbers. And All of those Natural Numbers will be starting with 1 so we will be having other Numbers as well on which No Number will be mapped like 2 or 211 or 79 So This Means Set of Natural Numbers is Grater then Real Numbers Between 0 and 1. So Set of Real Numbers Between 0 and 1 is Countably Infinite.

What's Ur Opinion ?

Answer 1

The set of real numbers between 0 and 1 is uncountably infinite, as shown by Cantor's diagonal argument which you are familiar with.

What may be surprising to you is that the set of rational numbers between 0 and 1 is countably infinite. That is, there is a 1-to-1 correspondence between the integers and all fractions and numbers with a finite decimal expansion. You can find the proof here.

Answer 2

This doesn't work because an arbitrary non-rational real number such as 0.5123129421... is a legitimate real number but the number 15123129421... isn't. in the case of the former, you can point out (at least in principle) where along the number line it would lie, but for the latter, it's impossible. Try to say out 15123129421... as one number (like 1022 is one thousand and twenty two). You won't be able to, because such number is not a natural number.