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Prove or falsify:

If the following limits exist: $$\lim_{n\to\infty}(a_n+b_n)$$ $$\lim_{n\to\infty}(a_n\cdot b_n)$$ Then the following limit exists: $$\lim_{n\to\infty}({a_n}^2+{b_n}^2)$$

Solution attempt: $$\lim_{n\to\infty}{{a_n}^2+{b_n}^2}=\lim_{n\to\infty}{{(a_n+b_n)}^2-2a_nb_n}=\lim_{n\to\infty}{a_n+b_n}\cdot\lim_{n\to\infty}{a_n+b_n}-2\lim_{n\to\infty}{a_nb_n}$$ So it converges too?

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  • $\begingroup$ You need to use the properties of the convergent sequences. $\endgroup$
    – njguliyev
    Aug 26, 2013 at 12:53
  • $\begingroup$ Your reasoning is correct; you are using that the product of two convergent sequences is convergent, and a constant times a convergent sequence is convergent. $\endgroup$
    – user84413
    Aug 26, 2013 at 18:45

1 Answer 1

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Hint: $x^2+y^2 = (x+y)^2-2xy$.

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