Informally speaking the sequence squareroot n converges to i

Informally speaking, the sequence squareroot n \"converges to infinity.\" Imitate the logical structure of Definition 2.2.3 to create a rigorous definition for the mathematical statement lima x_n = infinity. Use this definition to prove lim squareroot n = infinity. What does your definition in (a) say about the particular sequence (1,0,2,0,3,0,4,0,5,0,...)?

Solution

(a) let E >0 be given

let N be an integer such that N>E^2

=>

for n>=N we have n>E^2

=>

for n>=N we have sqrt(n)>E

=>

lim sqrt(n) = infinity

thus proved

(b)

x_2n-1 = n, x_2n = 0,

one subsequence is divergent (x2n) => given sequence is divergent