Suppose $b_n$ is a sequence $>0$ and $b>0$ where $b_n$ converge to $b$. Suppose $z_n=\log b_n$ and $z=\log b$, prove that $z_n$ converge to $z$. I know the definition of limit but not sure how to satisfy the condition
proving $z_n$ converge to $z$
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0In order to be able to help we have to know the rules of the game. What is your definition of $\log$, and what properties of $\log$ (or of sequences, for that matter) are you allowed to use? – 2012-09-22
3 Answers
Hint use $ \log \space a - \log\space b = \log \space a/b $ and $\log 1 = 0$
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0yup, but we have no idea even we got $b_n$ to find the $N$ from sequence {$z_n$} – 2012-09-22
ADD
I don't know if the hypothesis that the logarithm is continuous is the best choice, so I'll add something. From any definition of the logarithm, you'll extract that
$\log x - \log y = \log \frac{x}{y}$
and that
$1 - \frac{1}{x} < \log x < x - 1$
for $x\neq 1$. If $x=1$, we have equalities. From $(2)$, we have that, for $x\neq 1$,
$\frac{1}{x} < \frac{{\log x}}{{x - 1}} < 1$
From the squeeze theorem it follows that
$\mathop {\lim }\limits_{x \to 1} \frac{{\log x - \log 1}}{{x - 1}} = 1$
from where the logarithm is differentiable at $x=1$, and thus continuous at $x=1$. But the fact that it is continuous at $x=1$ means it is continuous for every $x>0$.
Indeed, pick any sequence $a_n>0$ that converges to $a(>0)$. Then
$\displaylines{ \mathop {\lim }\limits_{n \to \infty } \log {a_n} = \log a \cr \Leftrightarrow \mathop {\lim }\limits_{n \to \infty } \left( {\log {a_n} - \log a} \right) = 0 \cr \Leftrightarrow \mathop {\lim }\limits_{n \to \infty } \log \frac{{{a_n}}}{a} = 0 \cr} $
But $\frac {a_n}{a}\to 1$ and $\log 1=0$.
You have that $(b_n)$ is a sequence of positive numbers, that is, $b_n>0\;\forall n$, that converges to $b$. This means that for every $\epsilon>0$ there is an $N_0$ such that, whenever $n\geq N_0$, $|b-b_n|<\epsilon$.
Now, we're setting $z_n=\log \; b_n$. This makes sense for each $n$ for $b_n>0$. Now, we want to prove that, $z_n\to z=\log b$. This means that, for every $\epsilon >0$, there is an $N_1$ such that, whenever $n\geq N_1$, $|z_n-z|<\epsilon$, that is
$|\log b_n-\log b|<\epsilon$
But $\log x$ is continuous for $x>0$, this means that for any $\epsilon>0$ there is a $\delta >0$ such that, for all $x$,
$|x-a|<\delta\implies |\log x-\log b|<\epsilon$
But then, since $b_n$ converges to $b$, for any $\delta >0$, there will be an $N_\delta$ for which
$|b-b_n|<\delta$
and consequently
$|\log b_n-\log b|<\epsilon$
Thus, we can take $N=N_\delta$. This means that for any $\epsilon >0$, whenever $n\geq N_\delta$ we'll have $|z-z_n|=|\log b_n-\log b|<\epsilon$ as desired.
Setting $ x_n=\min\{b,b_n\},\ y_n=\max\{b,b_n\}, $ we have $ y_n+x_n=b+b_n,\ y_n-x_n=|b-b_n|. $ Therefore $ \lim_{n\to \infty}(y_n+x_n)=2b,\ \lim_{n\to \infty}(y_n-x_n)=0, $ and we deduce that $ \lim_{n\to \infty}y_n=\lim_{n\to \infty}x_n=b. $ Since $b>0$, we have $ \lim_{n \to \infty}\frac{y_n}{x_n}=1. $ Thus, given $\varepsilon>0$ there is an $N=N(\varepsilon) \in \mathbb{N}$ such that $ \frac{y_n}{x_n}-1\le \varepsilon \quad \forall\ n\ge N. $ Hence, for every $n \ge N$ we have $ |z-z_n|=\left|\int_b^{b_n}\frac{dt}{t}\right|=\int_{x_n}^{y_n}\frac{dt}{t}\le \frac{y_n-x_n}{x_n}=\frac{y_n}{x_n}-1 \le \varepsilon, $ i.e. $z_n \to z$ as $n \to \infty$.