用定义证明数列√(n+1)-√n的极限是0

Question

用定义证明数列√(n+1)-√n的极限是0

Answer 1

|√(n+1)-√n -0| < ε
|1/(√(n+1) + √n )| < ε
1/(2√n) < ε
n > { 1/(2ε) }^2
∀ε>0 ,∃N = [{ 1/(2ε) }^2] +1, st
|√(n+1)-√n -0| < ε , ∀N>n
=>
lim(n->∞) [√(n+1)-√n]=0