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Hi I attach my question.

In this I have assumed Archimeadan's property, there probably are other more efficient ways but I think what I have done is correct, if anyone could take a look at it- it would be awesome thnx

3 Answers 3

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It's better to look at these limits: $$ \lim_{n \to +\infty}{5(1+\frac{1}{n})}-\lim_{n \to +\infty}{2(1+\frac{1}{n^2})}= \lim_{n \to +\infty}{5}+\lim_{n \to +\infty}{\frac{5}{n}}-\lim_{n \to +\infty}{2}+\lim_{n \to +\infty}{\frac{2}{n^2}}=5+0-2 $$

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    Thank you for responding so quickly, this looks more cleaner than my method. – 2017-02-08
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No need for that. You know $\lim_{n\to\infty}\frac{1}n=0$ and $\lim_{n\to\infty}\frac{1}n\cdot \frac{1}n =$$\lim_{n\to\infty}\frac{1}n. \lim_{n\to\infty}\frac{1}n =0\cdot0=0$ Now use limit of sum is sum of limits for convergent sequences and you're done.

$\lim_{n\to\infty}5(1+\frac{1}n)- 2\cdot\lim_{n\to\infty}(1+\frac{1}n\cdot \frac{1}n)= 5-2=3$

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    Thank you too for your quick response, I had not thought to use tosolve this way, thank you :P – 2017-02-08
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    you're very welcome. – 2017-02-08
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This looks correct although the step where you remove the absolute value looks like you could add some more justification.

Some other pointers:

You could just say for N := $\lceil \frac{3}{\epsilon} \rceil$, then n>N implies $\frac{1}{n} < \frac{\epsilon}{3}$ to avoid referring to the property.

You could also pull a $\frac{1}{n}$ out of the original absolute value and have $\frac{1}{n}\left(5-\frac{2}{n}\right) < \frac{5}{n}$ at the end of your second line and just conclude the proof by setting N:=$\lceil\frac{5}{\epsilon}\rceil$

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    Yes thank you I was trying to follow the book's method that they implied which is why I negated the last line of your comment in my proof. – 2017-02-08
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    Can I please ask you that in the 4th line of your response, what does the colon before the equals sign mean - I see that used a lot. I suppose I could imply that for 5/ϵ as well. – 2017-02-08
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    It's used to mean "set a variable to a value". x=y is a statement that x is equal to y, whereas x:= y means set x to the value y. You'll sometimes see =: which means set the right side to the value from the left side. You'll sometimes see this in programming languages as well. Some languages use = to mean assign and == to be an equality test whereas others use := to assign to avoid ambiguity. – 2017-02-10
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    Thank you for this, I will upload an updated proof. Can I also ask you why you used square brackets, in this stage; \lceil \frac{3}{\epsilon} \rceil (in your 4th line) – 2017-02-10
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    I used ceilings instead of square brackets (note the lack of a foot on the bracket). Ceiling means "the smallest integer greater than or equal to this" whereas floor (like square brackets without the hat) means "the largest integer less than or equal to this". Basically, it just means round up to the next integer because N and n are usually assumed to be members of the Integers. – 2017-02-10