1. **Problem Statement:** We are given multiple questions about sets, sequences, functions, and series. We will solve the first question completely. 2. **a) Cardinalities of sets:** i) The set $\mathbb{Q}$ is the set of all rational numbers. It is countably infinite, so its cardinality is $\aleph_0$ (aleph-null). ii) The set $A = \{1,3,2,4,1,4,5,2,1,6\}$ has repeated elements. The distinct elements are $\{1,2,3,4,5,6\}$, so $|A| = 6$. iii) Given $A = \{a,b,c,d,e,f\}$ and $B = \{e,f,g,h\}$, the union is $A \cup B = \{a,b,c,d,e,f,g,h\}$. So $|A \cup B| = 8$. 3. **b) Definitions related to sequences:** i) A sequence $(x_n)$ converges to $L$ if for every $\varepsilon > 0$, there exists $N \in \mathbb{N}$ such that for all $n \geq N$, $|x_n - L| < \varepsilon$. ii) A sequence $(x_n)$ is bounded if there exists $M > 0$ such that for all $n$, $|x_n| \leq M$. iii) If $(x_n)$ is increasing and bounded above, then $(x_n)$ converges to its supremum. 4. **c) Convergence of sequences:** i) $(a_n) = \left(\frac{2n^5 + 4}{n^3 + 6}\right)$. For large $n$, dominant terms are $2n^5$ in numerator and $n^3$ in denominator. $$a_n \approx \frac{2n^5}{n^3} = 2n^2 \to \infty$$ So $(a_n)$ diverges to infinity. ii) $(x_n) = \left(-n + \sqrt{n^2 + 3n}\right)$. Rewrite: $$x_n = -n + n\sqrt{1 + \frac{3}{n}} = n\left(-1 + \sqrt{1 + \frac{3}{n}}\right)$$ Use binomial expansion for large $n$: $$\sqrt{1 + \frac{3}{n}} \approx 1 + \frac{3}{2n} - \frac{9}{8n^2}$$ So: $$x_n \approx n\left(-1 + 1 + \frac{3}{2n} - \frac{9}{8n^2}\right) = n\left(\frac{3}{2n} - \frac{9}{8n^2}\right) = \frac{3}{2} - \frac{9}{8n} \to \frac{3}{2}$$ So $(x_n)$ converges to $\frac{3}{2}$. 5. **d) General term of sequence $(2,5,8,11,14,...)$** This is an arithmetic sequence with first term $a_1=2$ and common difference $d=3$. Formula: $$a_n = a_1 + (n-1)d = 2 + 3(n-1) = 3n - 1$$ 6. **e) Uniform continuity of $f(x) = 5x + 8$ on $\mathbb{R}$** For any $\varepsilon > 0$, choose $\delta = \frac{\varepsilon}{5}$. Then for all $x,y \in \mathbb{R}$, $$|x - y| < \delta \implies |f(x) - f(y)| = |5x + 8 - (5y + 8)| = 5|x - y| < 5\delta = \varepsilon$$ Hence, $f$ is uniformly continuous on $\mathbb{R}$. 7. **f) Cauchy sequence and sequence $(x_n) = 1 + \frac{1}{5} + \frac{1}{5^2} + ... + \frac{1}{5^n}$** - A sequence $(x_n)$ is Cauchy if for every $\varepsilon > 0$, there exists $N$ such that for all $m,n \geq N$, $|x_n - x_m| < \varepsilon$. - The given sequence is a partial sum of a geometric series with ratio $r=\frac{1}{5}$. - Since $|r| < 1$, the series converges. - For $m > n$, $$|x_m - x_n| = \left|\sum_{k=n+1}^m \frac{1}{5^k}\right| \leq \sum_{k=n+1}^\infty \frac{1}{5^k} = \frac{1/5^{n+1}}{1 - 1/5} = \frac{1}{4 \cdot 5^n}$$ - Given any $\varepsilon > 0$, choose $N$ such that $\frac{1}{4 \cdot 5^N} < \varepsilon$. - Thus, $(x_n)$ is Cauchy. 8. **g) Partial sum and sum of telescoping series** $$\sum_{k=1}^\infty \frac{1}{k(k+1)}$$ Use partial fraction decomposition: $$\frac{1}{k(k+1)} = \frac{1}{k} - \frac{1}{k+1}$$ Partial sum: $$S_n = \sum_{k=1}^n \left(\frac{1}{k} - \frac{1}{k+1}\right) = 1 - \frac{1}{n+1}$$ Taking limit as $n \to \infty$: $$S = \lim_{n \to \infty} S_n = 1 - 0 = 1$$ 9. **h) Riemann integrability and integral of constant function** i) A bounded function $f$ on $[a,b]$ is Riemann integrable if the upper and lower sums can be made arbitrarily close, i.e., the set of discontinuities has measure zero. ii) For $f(x) = k$ constant, - $f$ is continuous on $[a,b]$, so Riemann integrable. - The integral is: $$\int_a^b k \, dx = k(b - a)$$ **Final answers:** i) $|\mathbb{Q}| = \aleph_0$ ii) $|A| = 6$ iii) $|A \cup B| = 8$ b) i) Convergence definition with $\varepsilon, N$ ii) Boundedness definition iii) Increasing and bounded above implies convergence c) i) Diverges to infinity ii) Converges to $\frac{3}{2}$ d) $a_n = 3n - 1$ e) $f$ is uniformly continuous on $\mathbb{R}$ f) $(x_n)$ is Cauchy g) $S_n = 1 - \frac{1}{n+1}$, $S = 1$ h) i) Riemann integrable if discontinuities measure zero ii) $\int_a^b k \, dx = k(b - a)$