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1. Proving b^{1/n} can be made arbitrarily close to 1

I am trying to prove that for any b \in \mathbb{R} such that b > 1. We have that for any n \in \mathbb{Z}^+, b^{\frac{1}{n}} can be made arbitrarily close to 1. I'm completing the routine outlined in Rudin's book. So I'm at the step where I need to prove: b - 1 \geq n(b^{\frac{1}{n}} - 1). And...
2. simple proof help (I think I'm really close)

Suppose that h_0, h_1, h_2, h_3, \dots is a sequence defined as follows: h_0 = 1, h_1 = 2, h_2 = 3, h_k = h_{k-1}+h_{k-2}+h_{k-3} for all ints k \geq 3. Prove that h_n\leq 3^n for all ints n \geq 0 What I have so far is: base case n=0, then h_0=1 and 1 \leq 3. inductive step: assume...
3. Definite Integral close??

Find the derivative of h(x) = integral(sin(x) at the top and -5 at the bottom) of cos(t^5) +t So... here's what I thought the answer should be, but I'm evidently missing something, again. My plan of attack is to insert sin(x) for t and then subtract the function with -5 inserted for t...
4. related rate angle so close...

A truck 20 feet high drives away from a video camera located on the ground at 10 feet per second. How fast is the angle between the road and the top of the truck shrinking from the point of view of the camera when the truck is 20 feet away? So far what I've gotten is that I can use tan and I...
5. Formal Modulus Proof: How close am I?

Show that if n is an odd positive integer then n^2 = 1(mod 8). I see that any odd square has 1 as a remainder when calculated. Example: 49 = 7 * 7 = 1(mod 4), and any odd number squared equals an odd number. Let 2k represent all positive even integers. So n^2 = 1(mod 2k) for all odd...
6. Compact and Close Set

Let K and F be nonempty subsets of R^n. Suppose that K is compact and F is closed. Define d = inf {||x-y|| : x in K, y in F}. Prove that d = 0 if and only if K intersect F = nonempty set.