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Find, with proof, the supremum of the subset $E$ of $\mathbb{R}$ defined by:

$E = \left\{\frac{25+16\sqrt{n}}{5+3\sqrt{n}}:n\in\mathbb{N}\right\}$

Is it ok to solve the limit of the function and call it a proof?

i.e. $$\text{sup}(E) = \lim_{n\to\infty} f(n) =\space ...$$

Or do I need to involve actual definitions of bounds etc.

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  • $\begingroup$ In general, limit may not give you the supremum. For example, consider $\left\{\frac{1}{n}\right\}$, the limit as $n \to \infty$ is $0$ but the supremum is $1$. $\endgroup$ – Anurag A Aug 13 at 23:31
  • $\begingroup$ $\frac{25+16\sqrt{n}}{5+3\sqrt{n}}=\frac{16}3-\frac5{15+9\sqrt{n}}$ $\endgroup$ – robjohn Aug 13 at 23:42
  • $\begingroup$ If you prove it is increasing you can do that. But you should cite the theorem that says you can do that. $\endgroup$ – fleablood Aug 13 at 23:42
  • $\begingroup$ Out of curiosity just how would you claim to limit? Whatever argument you use to declare a limit are basically the same, in this case, as declaring the supremum. $\endgroup$ – fleablood Aug 13 at 23:48
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You have an increasing sequence

$$ \left\{\frac{25+16\sqrt{n}}{5+3\sqrt{n}}:n\in\mathbb{N}\right\}$$

with the limit as $n$ goes to infinity equal to $16/3$

The limit is the supremum, that is $$\sup (E) = 16/3$$ All you have to do is to show that the sequence is increasing and bounded above by its limit.

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Hint As I pointed out in my comment, in general, the limit may not be the answer for supremum.

In this problem, observe that $$\frac{25+16\sqrt{n}}{5+3\sqrt{n}}=5+\frac{1}{3+\frac{5}{\sqrt{n}}}.$$ So the given sequence is an increasing sequence (you still need to show boundedness, which is easy). So the limit will, give you the supremum.

But simply computing the limit is not a proof that supremum is what you claim it to be.

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The supremum is the limit if the function in question is increasing. Doing some reduction shows

$$\frac{25+15\sqrt{n}}{5+3\sqrt{n}}=5+\frac{\sqrt{n}}{5+3\sqrt{n}}= 5+\frac{1}{ \frac{5}{\sqrt{n}}+3 }$$ which is clearly an increasing function of $n$. The supremum/limit of this function at natural number values is $5+1/3=16/3.$

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