# For a decimal expansion $K.d_1d_2d_3d_4\cdots$, prove $s_n < K + 1$ for all $n \in \mathbb N$

From the book Elementary Analysis: The Theory of Calculus by Kenneth A. Ross

I've been working on the exercises at the end of Chapter 2 $$\S$$ 10, but I just can't seem to figure this one out.

Exercise 10.3

For a decimal expansion $$K.d_1d_2d_3d_4\cdots$$, let $$(s_n)$$ be defined as in Discussion 10.3. Prove $$s_n < K + 1$$ for all $$n \in \mathbb N$$. Hint: $$\frac{9}{10} + \frac{9}{10^2} + \cdots + \frac{9}{10^n} = 1 - \frac{1}{10^n}$$.

Here is Discussion 10.3:

10.3 Discussion of Decimals

We restrict our attention to nonnegative decimal expansions and nonnegative real numbers. From our point of view, every nonnegative decimal expansion is shorthand for the limit of a bounded increasing sequence of real numbers. Suppose we are given a decimal expansion $$K.d_1d_2d_3d_4\cdots$$, where $$K$$ is a nonnegative integer and each $$d_j$$ belongs to $$\{0,1,2,3,4,5,6,7,8,9\}$$. Let

$$s_n = K + \frac{d_1}{10} + \frac{d_2}{10^2} + \cdots + \frac{d_n}{10^n}\tag{1}\label{1}.$$

Then $$(s_n)$$ is an increasing sequence of real numbers, and $$(s_n)$$ is bounded [by $$K + 1$$, in fact]. So by Theorem 10.2, $$(s_n)$$ converges to a real number we traditionally write as $$K.d_1d_2d_3d_4\cdots$$. For example, $$3.3333\cdots$$ represents

$$\lim_{n\to\infty} \left(3 + \frac{3}{10} + \frac {3}{10^2}\ + \cdots + \frac{3}{10^n}\right).$$

To calculate this limit, we borrow the following fact about geometric series:

$$\lim_{n\to\infty} a\left(1 + r + r^2 + \cdots + r^n\right) = \frac{a}{1-r} \quad \text{for} \quad |r| < 1 \tag{2}\label{2}.$$

In our case, $$a = 3$$ and $$r = \frac{1}{10}$$, so $$3.3333\cdots$$ represents $$\frac{3}{1-\frac{1}{10}}=\frac{10}{3}$$, as expected. Similarly, $$0.9999\cdots$$ represents

$$\lim_{n\to\infty} \left(\frac{9}{10} + \frac{9}{10^2} + \cdots + \frac{9}{10^n}\right) = \frac{\frac{9}{10}}{1-\frac{1}{10}} = 1.$$

Thus $$0.9999\cdots$$ and $$1.0000\cdots$$ are different decimal expansions that represent the same number.

This section in the book proved two important theorems (and the related lemmas/corollaries): (1) All bounded monotone sequences converge; and (2) A sequence is a convergent sequence if and only if it is a Cauchy sequence. So I'm wondering how I would use those theorems in this proof? Or would I? Would I have to use the ideas of lim sup and lim inf? Geometric series?

$s_{n} < K +1 \leftrightarrow K + d_{1}/10 + d_{2}/10 + ... d_{n}/10 < K +1 \leftrightarrow d_{1}/10 + d_{2}/10 + ... d_{n}/10 < 1 \leftarrow (by \quad hint) 1- 1/10^{n} < 1 \leftrightarrow 0 < 1/10^{n}$ with the last statement being trivially true. We can use the hint as taking all the $d_{j}$ values to be 9 is the biggest $s_{n}$ can be.