# Thread: Couple of similar proofs

1. ## Couple of similar proofs

Let x1,x2,..., xn be positive real numbers. Prove that

n^2 <= (x1 + x2 + ... + xn ) ( 1/x1 + 1/x2 + ... + 1/xn )

and

(x1 + x2 + ... + xn)/(sqrt(n)) <= sqrt( x1^2 + x2^2 +...+ xn^2)

2. Still not able to solve the second one, any help appreciated.

I was able to solve the first one using induction and the lemma from another thread i asked about that (x/y) + (y/x) > 2

Using induction, you can factor the sequence of k + 1 in to the sequence up to k, and (x/y) + (y/x) for each where x = x1, x2,...,xk and y = x(k+1). Since there are k of those factors and each is greater than 2, we have the first two factors are great than k^2 + 2k. Also, when factoring you get a factor of 2, so with K^2 + 2k + 2 > K^2 + 2k + 1 = (K+1)^2, the theorem is proved by induction.

3. Nvm, the second question is quite easy.

Let a_n = x1 + x2 + ... + xn
let b_n = 1 + 1 + ... + 1 = n

Then the inequality holds from the Cauchy-Swartz inequality using a little simplification