# Thread: question in sup

1. ## question in sup

i need to prove that
sup ${(xsin(x))/(x+1) |x>0}}=1$

i succeded proving that 1 is an upper bound of $f(x)$
now i need to prove that 1 is the lowest upper bound. i tried to assume that there's another number (let called $c$), that is also an upper bound of $f(x)$ but is lower than 1.
$c<1$
but i don't really know how to go forward..
what can i do? i'm so suck in proving things...

2. ## Re: question in sup

Hang on, I can barely read this, are you asking to prove that the supremum of $\displaystyle \frac{\sin{(x)}}{x} + 1$ is 0 if x > 0?

3. ## Re: question in sup

sorry, i don't know how to add math to a post..
i ment the supremum of (xsinx)/(x+1)=1 (at x>0)

4. ## Re: question in sup

That can be written $\frac{x}{x+1}sin(x)$. Now look at $x= N\pi/2$ for N a very large odd number.

5. ## Re: question in sup

Originally Posted by HallsofIvy
That can be written $\frac{x}{x+1}sin(x)$. Now look at $x= N\pi/2$ for N a very large odd number.
yeah, in this condition (N approaches to infinity) it will all approach 1. but is it just enough - does it necessarly tell me that there's no other upper bound of $f(x)$ which is smaller than 1??

6. ## Re: question in sup

(i accidently posted another post, and i can't delete it..)

7. ## Re: question in sup

Originally Posted by orir
yeah, in this condition (N approaches to infinity) it will all approach 1. but is it just enough - does it necessarly tell me that there's no other upper bound of $f(x)$ which is smaller than 1??

Note that ${\lim _{x \to \infty }}\frac{x}{{x + 1}} = 1$

Also if $n\in\mathbb{Z}^+$ then $\sin \left( {\frac{{(1 + 4n)\pi }}{2}} \right) = 1$.

If you put those two facts together, then it easy to see why $1\text{ is the }\sup$.

8. ## Re: question in sup

i'm sorry, but it's not easy to me to see from that that 1 is the sup. can you explain more? (and why 1+4n?)

9. ## Re: question in sup

Originally Posted by orir
i'm sorry, but it's not easy to me to see from that that 1 is the sup. can you explain more? (and why 1+4n?)
No I cannot teach you basic mathematics.

10. ## Re: question in sup

Originally Posted by Plato
No I cannot teach you basic mathematics.
i didn't ask you to teach me basic math, so you don't need to be so supercilious.. just wanted a bit more explanation, that's it.

11. ## Re: question in sup

Plato is pointing out that the sine function can never be any greater than 1. Since the two functions are always bounded above by 1, then so must be their product...