Induction proofs of inequalities

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Problem 6 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 1.1 "Real Numbers"

Prove Bernoulli's inequality

(1+x1)(1+x2)⋯(1+xn)≥1+x1+x2+⋯+xn. (1+x_{1})(1+x_{2})\cdots (1+x_{n}) \geq 1+x_{1}+x_{2}+\cdots +x_{n} .

where x1,x2,…,xnx_{1}, x_{2}, \ldots , x_{n} are numbers with the same sign, all greater than −1-1.

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Problem 8 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 1.1 "Real Numbers"

Prove the inequality

n!<(n+12)nfor n>1. n! < \left(\frac{n+1}{2}\right)^{n} \quad \text{for } n>1 .
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Problem 9 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 1.1 "Real Numbers"

Prove the inequality

2!⋅4!⋯(2n)!>[(n+1)!]nfor n>1. 2! \cdot 4! \cdots (2n)! > \left[(n+1)!\right]^{n} \quad \text{for } n>1 .
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Problem 10 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 1.1 "Real Numbers"

Prove the inequality

12⋅34⋯2n−12n<12n+1. \frac{1}{2} \cdot \frac{3}{4} \cdots \frac{2n-1}{2n} < \frac{1}{\sqrt{2n+1}} .
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Problem 10.1 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Problem 10 "[NO TITLE]"
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Problem 1.2.9 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.2.2 "Problems"

Prove that for every n∈Nn \in \mathbb {N} Bernoulli's inequality (1+a)n⩾1+na(1+a)^{n} \geqslant 1 + na holds, if a>−1a > -1.

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Problem 1.2.10 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.2.2 "Problems"

Prove that for every n∈Nn \in \mathbb {N} the following inequality holds:

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(1)

1n+1+1n+2+…+13n+1>1\frac{1}{n+1}+\frac{1}{n+2}+\ldots +\frac{1}{3 n+1} > 1;

(2)

1n+1+1n+2+…+12n>1324\frac{1}{n+1}+\frac{1}{n+2}+\ldots +\frac{1}{2 n} > \frac{13}{24};

(3)

12⋅34⋅56⋅…⋅2n−12n<13n+1\frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdot \ldots \cdot \frac{2 n-1}{2 n} < \frac{1}{\sqrt{3 n+1}};

(4)

n<1+12+…+1n<2n\sqrt{n} < 1+\frac{1}{\sqrt{2}}+\ldots +\frac{1}{\sqrt{n}} < 2 \sqrt{n}.

Problem 1.2.12 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.2.2 "Problems"

Let x1,x2,…,xnx_{1}, x_{2}, \ldots , x_{n} be arbitrary positive numbers such that x1x2…xn=1x_{1} x_{2} \ldots x_{n} = 1. Prove that

x1+x2+…+xn⩾n x_{1}+x_{2}+\ldots +x_{n} \geqslant n
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Problem 1.2.13 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.2.2 "Problems"

Let x1,x2,…,xnx_{1}, x_{2}, \ldots , x_{n} be arbitrary positive numbers. Prove that

x1+x2+…+xnn⩾x1x2…xnn \frac{x_{1}+x_{2}+\ldots +x_{n}}{n} \geqslant \sqrt[n]{x_{1} x_{2} \ldots x_{n}}

where equality holds if and only if x1=x2=…=xnx_{1} = x_{2} = \ldots = x_{n}.

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Problem 1.7.253 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.7.2 "Problems"

Let {xnk}\left\{ x_{n_{k}}\right\} be a subsequence of the sequence {xn}\left\{ x_{n}\right\}. Prove that nk⩾k,k∈Nn_{k} \geqslant k, k \in \mathbb {N}.

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Problem I.2.4 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section I.2 "Inequalities"

Let {qk}k=1n\left\{ q_{k}\right\}_{k=1}^{n} be a finite sequence of numbers, with qk⩾1q_{k} \geqslant 1 for all kk. Prove the inequality

1+1+…+1qnq2q1⩽1+1q1+1q1q2+…+1q1q2…qn. \sqrt[q_{1}]{1+\sqrt[q_{2}]{1+\ldots +\sqrt[q_{n}]{1}}} \leqslant 1+\frac{1}{q_{1}}+\frac{1}{q_{1} q_{2}}+\ldots +\frac{1}{q_{1} q_{2} \ldots q_{n}} .
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Problem 1.1.12 from Rudin W. "Principles of Mathematical Analysis" [Rud76]More problems in Chapter 1.1 "The Real and Complex Number Systems"

If z1,…,znz_{1}, \ldots , z_{n} are complex, prove that

∣z1+z2+⋯+zn∣≤∣z1∣+∣z2∣+⋯+∣zn∣ \left|z_{1}+z_{2}+\cdots +z_{n}\right| \leq \left|z_{1}\right|+\left|z_{2}\right|+\cdots +\left|z_{n}\right|
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Problem 1.8.8 from Rudin W. "Principles of Mathematical Analysis" [Rud76]More problems in Chapter 1.8 "Some Special Functions"

For n=0,1,2,…n = 0,1,2, \ldots, and xx real, prove that

∣sin⁡nx∣≤n∣sin⁡x∣ \left|\sin n x\right| \leq n\left|\sin x\right|

Note that this inequality may be false for other values of nn. For instance,

∣sin⁡12π∣>12∣sin⁡π∣ \left|\sin \frac{1}{2} \pi \right| > \frac{1}{2}\left|\sin \pi \right|
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Problem 4.6 from Кострикин А.И. "Сборник задач по алгебре" [Кос15]More problems in Section 4 "Recurrence relations. Induction"

Let a⩾−1a \geqslant -1. Prove that for any natural number nn the inequality (1+a)n⩾1+na(1+a)^{n} \geqslant 1+n a holds.

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Problem 4.7 from Кострикин А.И. "Сборник задач по алгебре" [Кос15]More problems in Section 4 "Recurrence relations. Induction"

Prove that for any natural number n⩾2n \geqslant 2 the inequality holds

4nn+1<(2n)!(n!)2 \frac{4^{n}}{n+1}<\frac{(2 n)!}{(n!)^{2}}
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Problem 4.10 from Кострикин А.И. "Сборник задач по алгебре" [Кос15]More problems in Section 4 "Recurrence relations. Induction"

Prove that for any natural number nn the inequalities hold

n2<1+12+13+…+12n−1⩽n \frac{n}{2}<1+\frac{1}{2}+\frac{1}{3}+\ldots +\frac{1}{2^{n}-1} \leqslant n
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Problem 41 from Дехтярь М.И., Дудаков С.М., Карлов М.И., Карлов Б.Н. "Задачник по дискретной математике" [ДДК+21]More problems in Chapter 2 "The Method of Mathematical Induction"

Using induction, prove that

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(a)

2n+1⩽2n2 n+1 \leqslant 2^{n} for n⩾3n \geqslant 3;

(b)

n2⩽2nn^{2} \leqslant 2^{n} for n⩾4n \geqslant 4.

References

[Dem97] Demidovich B.P. Сборник задач и упражнений по математическому анализу. 13-е изд., Издательство Московского университета / Издательство "ЧеРо", 1997.
[ККЧ+03] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Предел. Непрерывность. Дифференцируемость. 2-е изд., ФИЗМАТЛИТ, 2003.
[МГЛ+04] Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. Избранные задачи по вещественному анализу. 2-е изд., Невский Диалект; БХВ-Петербург, 2004.
[Rud76] Rudin W. Principles of Mathematical Analysis. 3rd ed., McGraw-Hill, 1976.
[Кос15] Кострикин А.И. Сборник задач по алгебре. МЦНМО, 2015.
[ДДК+21] Дехтярь М.И., Дудаков С.М., Карлов М.И., Карлов Б.Н. Задачник по дискретной математике. 2-е изд., Тверской государственный университет, 2021.