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I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).
I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...
I need help with a part of Exercise 2.2.7 Part (1) ... ...
Exercise 2.2.7 Part (1) reads as follows:View attachment 7189I have managed a solution to this exercise and am posting it because
(1) I am unsure that the proof is valid/correct ... particularly given my arithmetic on \epsilon ...
(2) I did not use Sohrab's hint ... and that concerns me ... but I am also interested in how Bernoulli's Inequality is used in this proof ... I can see how to simply involve Bernoulli's Inequality, but not how to use it profitably ...
My proof is as follows:
To show $$\text{lim } ( \frac{ 1 }{ 1 + na } ) = 0$$ where $$n \in \mathbb{N} $$ ...
We have $$| \frac{ 1 }{ 1 + na } - 0 | = \frac{ 1 }{ 1 + na } \lt \frac{ 1 }{ na }$$
Now for $$\epsilon \gt 0$$ let $$\frac{ 1 }{ na } \lt \epsilon$$
Then let $$\epsilon = \frac{\epsilon^*}{a}$$ so that $$\epsilon^* = a \epsilon$$ ...
Then we have :
$$\frac{ 1 }{ 1 + na } \lt \frac{\epsilon^*}{a} \Longrightarrow \frac{ 1 }{ n } \lt \epsilon^*$$ ...So ... choose $$N$$ so that $$\frac{ 1 }{ N } \lt \epsilon^*$$ ...
So that then we have ... for $$n \gt N$$ ...
$$| \frac{ 1 }{ 1 + na } - 0 | = \frac{ 1 }{ 1 + na } \lt \frac{ 1 }{ na } = ( \frac{ 1 }{ n } ) ( \frac{ 1 }{ a } ) \lt \epsilon^* ( \frac{ 1 }{ a } ) = \epsilon a ( \frac{ 1 }{ a } ) = \epsilon $$
Is that correct?
and further ...
Can someone please indicate how Bernoulli's Inequality is used in a proof ...
Peter==========================================================================================
To help readers understand Sohrab's notation I am posting his definitions of convergence of a sequence and the limit of a sequence ... as follows:View attachment 7190
I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...
I need help with a part of Exercise 2.2.7 Part (1) ... ...
Exercise 2.2.7 Part (1) reads as follows:View attachment 7189I have managed a solution to this exercise and am posting it because
(1) I am unsure that the proof is valid/correct ... particularly given my arithmetic on \epsilon ...
(2) I did not use Sohrab's hint ... and that concerns me ... but I am also interested in how Bernoulli's Inequality is used in this proof ... I can see how to simply involve Bernoulli's Inequality, but not how to use it profitably ...
My proof is as follows:
To show $$\text{lim } ( \frac{ 1 }{ 1 + na } ) = 0$$ where $$n \in \mathbb{N} $$ ...
We have $$| \frac{ 1 }{ 1 + na } - 0 | = \frac{ 1 }{ 1 + na } \lt \frac{ 1 }{ na }$$
Now for $$\epsilon \gt 0$$ let $$\frac{ 1 }{ na } \lt \epsilon$$
Then let $$\epsilon = \frac{\epsilon^*}{a}$$ so that $$\epsilon^* = a \epsilon$$ ...
Then we have :
$$\frac{ 1 }{ 1 + na } \lt \frac{\epsilon^*}{a} \Longrightarrow \frac{ 1 }{ n } \lt \epsilon^*$$ ...So ... choose $$N$$ so that $$\frac{ 1 }{ N } \lt \epsilon^*$$ ...
So that then we have ... for $$n \gt N$$ ...
$$| \frac{ 1 }{ 1 + na } - 0 | = \frac{ 1 }{ 1 + na } \lt \frac{ 1 }{ na } = ( \frac{ 1 }{ n } ) ( \frac{ 1 }{ a } ) \lt \epsilon^* ( \frac{ 1 }{ a } ) = \epsilon a ( \frac{ 1 }{ a } ) = \epsilon $$
Is that correct?
and further ...
Can someone please indicate how Bernoulli's Inequality is used in a proof ...
Peter==========================================================================================
To help readers understand Sohrab's notation I am posting his definitions of convergence of a sequence and the limit of a sequence ... as follows:View attachment 7190