Statement 1 | Suppose {s_n} and {t_n} are sequences of complex numbers such that lim_n→∞ s_n = s and lim_n→∞ tn = t. lim_n→∞ s_nt_n = st. Statement 2 | There exists a bounded sequence of real numbers with no convergent subsequence.

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∻Solution by Steps∻ ‖ step 1 ⋮ To find the definite integral of $ x^4 $ from $ x = 3 $ to $ x = 5 $, we use the power rule for integration. ‖ ‖ step 2 ⋮ The power rule for integration states that the integral of $ x^n $ is $ \frac{x^{n+1}}{n+1} $ plus a constant of integration. ‖ ‖ step 3 ⋮ Applying the power rule to $ x^4 $, we get $ \frac{x^5}{5} $. ‖ ‖ step 4 ⋮ We evaluate this antiderivative from $ x = 3 $ to $ x = 5 $. ‖ ‖ step 5 ⋮ The definite integral is $ \left[\frac{x^5}{5}\right]_3^5 = \frac{5^5}{5} - \frac{3^5}{5} $. ‖ ‖ step 6 ⋮ Calculating the values, we get $ \frac{3125}{5} - \frac{243}{5} = \frac{2882}{5} $. ‖ ‖ step 7 ⋮ The result is $ \frac{2882}{5} $ or $ 576.4 $ in decimal form. ‖ ∻Answer∻ ⚹ $ \frac{2882}{5} $ or $ 576.4 $ ⚹ ∻Key Concept∻ ⚹Definite Integral and Power Rule for Integration⚹ ∻Explanation∻ ⚹The definite integral of a power function $ x^n $ from $ a $ to $ b $ is found using the power rule, which involves finding the antiderivative and evaluating it at the upper and lower limits of integration. The result is the net area under the curve from $ a $ to $ b $.⚹ Regarding the student's question about sequences: ∻Solution by Steps∻ ‖ step 1 ⋮ Statement 1 is a property of limits of sequences of complex numbers. ‖ ‖ step 2 ⋮ If $ \lim_{n \to \infty} s_n = s $ and $ \lim_{n \to \infty} t_n = t $, then $ \lim_{n \to \infty} (s_n t_n) = st $. ‖ ‖ step 3 ⋮ This follows from the limit laws for sequences, which state that the limit of a product is the product of the limits, provided that both limits exist. ‖ ∻Answer∻ ⚹ True ⚹ ‖ step 1 ⋮ Statement 2 is about bounded sequences of real numbers. ‖ ‖ step 2 ⋮ A bounded sequence is one where all terms of the sequence are within some fixed distance from a particular point. ‖ ‖ step 3 ⋮ The Bolzano-Weierstrass theorem states that every bounded sequence in $\mathbb{R}^n$ has a convergent subsequence. ‖ ‖ step 4 ⋮ Therefore, it is not possible for a bounded sequence of real numbers to have no convergent subsequence. ‖ ∻Answer∻ ⚹ False ⚹ ∻Key Concept∻ ⚹Limits of Sequences and Bolzano-Weierstrass Theorem⚹ ∻Explanation∻ ⚹Statement 1 is true due to the limit laws for sequences. Statement 2 is false because the Bolzano-Weierstrass theorem guarantees that every bounded sequence of real numbers has at least one convergent subsequence.⚹◊What is the definition of a bounded sequence of real numbers with no convergent subsequence in mathematics?⍭ Generate me a similar question◊

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