Constructing a Function F: Natural Restriction Homework

[benf.stokes]

Homework Statement

Let (xn) n ∈ N, be a succession. Construct a function F: [0,plus infinity[ to Rn (if possible
continuous) whose restriction to the naturals is (xn) n ∈ N, ie xn = F (n) for all n ∈ N, and such that ∃ limx → + ∞ F (x) = L if and only if ∃ Limn → + ∞ xn = L

The Attempt at a Solution

What does restriction to the naturals mean? Does it mean that F(n) must be a natural? If so I can't find out what function has domain [0,plus infinity[ and produces naturals.

Thanks

 

[HallsofIvy]

Homework Statement

Let (xn) n ∈ N, be a succession. Construct a function F: [0,plus infinity[ to Rn (if possible
continuous) whose restriction to the naturals is (xn) n ∈ N, ie xn = F (n) for all n ∈ N, and such that ∃ limx → + ∞ F (x) = L if and only if ∃ Limn → + ∞ xn = L

The Attempt at a Solution

What does restriction to the naturals mean? Does it mean that F(n) must a natural? If so I can't find out what function has domain [0,plus infinity[ and produces naturals.

Thanks

No, it is the domain, not the range, that is restricted. You are looking for a function, F(x), (where x can be any non-negative real number) such that for x= n, a non-negative integer, $F(x)= F(n)= x_n$.

For example, if $x_n= 2/n$, then you would take F(x)= 2/x.

 

[benf.stokes]

Thanks for the reply. Would F(x) = sqrt(x)/(x^2+1) be an acceptable function then?

 

[HallsofIvy]

Looks to me like it will work. What is $\lim_{x\to\infty}\sqrt{x}/(x^2+ 1)$?
What is $\lim_{n\to\infty}\sqrt(n)/(n^2+ 1)$?