Answer :

The statement to be proved is that if aₙ ≥ 0 for all n and limₙ→∞ aₙ = a, then limₙ→∞ √aₙ = √a.

To prove this, we can use the definition of a limit. Let's assume ε > 0 be given. Since limₙ→∞ aₙ = a, there exists an integer N such that for all n ≥ N, |aₙ - a| < ε.

Now, let's consider the sequence √aₙ. We want to show that limₙ→∞ √aₙ = √a.

For all n ≥ N, we have |√aₙ - √a| = |√aₙ - √a| * |√aₙ + √a| / |√aₙ + √a| (multiplying and dividing by the conjugate).

Using the difference of squares, we can simplify this to |√aₙ - √a| = |aₙ - a| / |√aₙ + √a|.

Since aₙ ≥ 0 for all n, and √aₙ ≥ 0, we have |√aₙ - √a| = |aₙ - a| / (√aₙ + √a) ≤ |aₙ - a| / (√a + √a).

By choosing N such that |aₙ - a| < ε, we can ensure that |√aₙ - √a| < ε / (√a + √a).

Therefore, as n approaches infinity, √aₙ approaches √a, and the limit limₙ→∞ √aₙ = √a holds.

Hence, we have proved that if aₙ ≥ 0 for all n and limₙ→∞ aₙ = a, then limₙ→∞ √aₙ = √a.

Learn more about limit visit:

brainly.com/question/12211820

#SPJ11

Thanks for taking the time to read 14 If an 0 for all n and lim as n infinity an a prove that lim as n infinity of sqr an sqr a. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada