Arithmetic operations on sequences

[Grant Bonner]

If {a_n} converges to a with a_n>=0 for all n, show {sqrt(a_n)} converges to sqrt(a).

If a>0, then sqrt(a_n) - sqrt(a) = (a_n - a)/(sqrt(a_n) + sqrt(a)) <= a_n - a < e. Is that correct?

 

[daon]

For one thing, youre assuming that sqrt(a_n)-sqrt(a) > 0. Can't do that.

Secondly you're assuming that sqrt(a)+sqrt(a_n) >= 1 in your second-to-last inequality. Can't do that, either.

Start with the asusmption that for all e > 0 you can find an N s.t. n>N implies |a_m-a|<e.

You want to show for all e>0 you can find an M such that m>M implies |sqrt(a_m)-sqrt(a)|<e.