Proof for Convergence of Nested Sequences

Question

With this being my first week learning about limits, their rules, and convergence/divergence, I received another question:

$a_k$ is a convergent sequence with $\lim_{n\to \infty}(a_k) = a$

Let $p(x) = 3x^2 - x + 2$. Show that $(p(a_k))$ is also convergent with $\lim_{k\to \infty}p(a_k) = p(a)$

My approach would be to use the multiplication rule, which is: when you have two convergent sequences $\lim_{n\to \infty}(a_n) = a$ and $\lim_{n\to \infty}(b_n) = b$, then $\lim_{n\to \infty}(a_n \cdot b_n) = \lim_{n\to \infty}(a_n) \cdot \lim_{n\to \infty}(b_n) = a \cdot b$

But I'm not entirely sure about how to apply that rule specifically to this question. I get confused when so many variables are used. I also thought about manually listing out both sequences and looking at $a_k$ as a subsequence to $p(x)$, yet I'm not certain if that'd work.

I hope someone can clarify the best approach to both parts and explain their answer in detail. Thank you!!

Answer 1

Have you gone over the definition of continuity for real valued functions in your course yet? If so you can use the continuity of the polynomials in (a) and (b) to prove the result.

Otherwise your approach for using sums and multiplication of sequences will work. Indeed you can rewrite your sequence $p(a_k)$ as $p(a_k) = 3 \times a_k \times a_k -a_k +2$.

From here can you see where adding and multiplying sequences will give you what you want?

Answer 2

You are on the right track. In part a) $$p(a_k)= 3a_k^2-a_k+2$$

Now you need to use various rules, like the product rule and the sum rule. For the first term, $$\lim_{k\to\infty}a_k^2=\lim_{k\to\infty}a_k\cdot\lim_{k\to\infty}a_k=a^2$$

Once you complete part a) you should find part b) simple.