u/Sea_Handle_994

In Analysis I by Terence Tao, Tao rigorously develops the real number system and proves basic results of real anlysis. In Appendix B, he formally defines the decimal system. Proposition B.2.3 in particular reads:

>Proposition B.2.3 (Failure of uniqueness of decimal representations). The number 1 has two different decimal representations: 1.000... and 0.999....

>Proof. The representation 1 = 1.000... is clear. Now let’s compute 0.999.... By definition, this is the limit of the Cauchy sequence

>0.9, 0.99, 0.999, 0.9999, ... .

>But this sequence has 1 as a formal limit by Proposition 5.2.8.

Now that this rookie mistake by Terence Tao has finally been exposed, should I throw this textbook in the trash?