Discrete Math: Proving a Homework Statement

[planauts]

Homework Statement

http://puu.sh/1OfE2

Homework Equations

The Attempt at a Solution

I am not really sure about this one! :(
I think it's 1 because
http://puu.sh/1OfY0
http://puu.sh/1OfYE

I came up the number by working backwards (assuming the conclusion is true). However, for a proof, I cannot assume the conclusion is true and try proving the hypothesis. Could someone nudge me to the right direction in proving this statement?

Thanks!

 

[CompuChip]

As you say,
$$\lim_{x \to \infty} 2^{\frac{1}{x}} = 1$$
But since this is discrete mathematics, perhaps it's more intuitive to define a_n = 2^1/n and write
$$\lim_{n \to \infty} a_n = 1$$

Now can you solve it if I say: "$r = 1 + \epsilon$" and "definition of limit"?

 

[planauts]

No, I don't understand the second part with epsilon.

 

[CompuChip]

Do you know what the definition of the limit is?

 

[lurflurf]

Consider n<0

 

[planauts]

When I graphed it, i found out that r > 0.5 because 2^(-1) is 0.5 since n has to be int and -1 is an int but i don't know how to prove it. Graphing is not a good way, according to my Prof.

 

[CompuChip]

If you haven't learned the definition of limit yet, another approach is as follows: try solving the equation 2^(1/x) = r first. Once you find x for which the equality holds, you can may use your graph for inspiration for an integer n such that the inequality holds.