Proof by Contradiction: 0.999… = 1

In the previous post, we proved that

$latex \displaystyle a < \frac{a + b}{2}$

both by direct and indirect proof. The proof that we have used in the previous post was proof by contradiction. In proof by contradiction, we show that by assuming the proposition to false would imply a contradiction.  One of the most famous of this proof is the proof that there are infinitely many prime numbers.

In this post, we will prove that $latex 0.999 \cdots 1$. Recall that we already proved this theorem using direct proof.  In proving by contradiction, first, we assume that

$latex 0.999 \cdots < 1$

then find a contradiction somewhere. We then conclude that our assumption cannot be be false.  Continue reading

Proof that 0.999… = 1


There are certain concepts in mathematics that are counterintuitive. In this post, we discuss one of these concepts — the elementary proof that $latex 0.999 \cdots = 1$. The $latex \cdots$ symbol indicates that there are infinitely many $latex 9$’s to the right hand side of the decimal point. The proof of this theorem is extremely easy;  however, to be able to appreciate it will require understanding of the concept of limits and infinity.


Theorem: $latex 0.999 \cdots = 1$

Proof 1  Continue reading