Expert: Paul Klarreich - 3/7/2009

Question
Let x_n >= 0 for all n in the natural numbers.
1. If (x_n) converges to 0, show that (sqrt(x_n)) converges to 0.
2. If (x_n) converges to x, show that (sqrt(x_n)) converges to sqrt(x)

Please help!

Answer
Questioner:   Katy
Category:  Advanced Math
Private:  No
 
Subject:  real analysis sequences
Question:  Let x_n >= 0 for all n in the natural numbers.
1. If (x_n) converges to 0, show that (sqrt(x_n)) converges to 0.
2. If (x_n) converges to x, show that (sqrt(x_n)) converges to sqrt(x)

Please help!
............................
One of the rules for asking questions is:  TELL ME JUST WHAT YOU ARE STUDYING, SO I KNOW WHAT NOT TO USE.

For example:

sqrt(x) is continuous.  therefore:

lim(x[n] -> 0) sqrt(x[n]) = sqrt(0).

and

lim(x[n] -> x) sqrt(x[n]) = sqrt(x).

But perhaps you are not allowed to use continuity;  if not, let me know and I will try some other approach.