Show that      (1)  \(\displaystyle a_{n}\to a \iff (a_{n}-a)\to 0\),      (2)  \(\displaystyle a_{n}\to 0 \iff |a_{n}|\to 0\).

Difficulty: 1.

Show that \(\displaystyle \lim_{n\to\infty}a_{n}=\infty\iff\forall K\in \R\) only finitely many members of \(\displaystyle (a_{n})\) are smaller than \(\displaystyle K\).

Difficulty: 1.

Show that if \(\displaystyle a_{n}\to 0\) and \(\displaystyle a_{n}\not=0\), then \(\displaystyle \frac{1}{|a_{n}|}\to\infty\).

Difficulty: 1.

\(\displaystyle 0<a_n<1\) for all \(\displaystyle n \in \N\). Does it imply that \(\displaystyle a_n^n\to 0\)?

Difficulty: 2.

Suppose that \(\displaystyle a_{2n}\to B\), \(\displaystyle a_{2n+1}\to B\). Does it imply that \(\displaystyle a_n\to B\)?

Difficulty: 2.

Let \(\displaystyle a_n\) be a sequence of real numbers. Write down the negation of the statement \(\displaystyle \lim a_n=7\) (do not start with negation!).

Difficulty: 2.

Prove that \(\displaystyle a_n\to\infty\) implies that \(\displaystyle \sqrt{a_n}\to\infty\).

Difficulty: 2.

Is it true that if \(\displaystyle x_n\) is convergent, \(\displaystyle y_n\) is divergent then \(\displaystyle x_ny_n\) is divergent?

Difficulty: 2. Solution is available for this problem.

Show that if \(\displaystyle \forall n\geq n_{0}\) \(\displaystyle a_{n}\leq b_{n}\) and \(\displaystyle a_{n}\to\infty\), then \(\displaystyle b_{n}\to\infty.\)

Difficulty: 2.

Prove that if \(\displaystyle (a_n)\) is convergent, then \(\displaystyle (|a_n|)\) is convergent, too. Does the reverse implication also hold?

Difficulty: 2.

Suppose \(\displaystyle 0<a_n\to 0\). Prove that there are infinitely many \(\displaystyle n\) for which \(\displaystyle a_n>a_{n+r}\) for all \(\displaystyle r=1,2,\ldots\).

Difficulty: 3.

Does \(\displaystyle {a_n \over 3-a_n}\to 2\), imply \(\displaystyle a_n\to 2\)?

Difficulty: 3.

Prove that \(\displaystyle x_n\to a\not=0\) implies \(\displaystyle \lim {x_{n+1}\over x_n}=1\).

Difficulty: 3.

Prove that \(\displaystyle a_n\ge0\) and \(\displaystyle a_n\to a\) implies \(\displaystyle \sqrt{a_n}\to\sqrt{a}\).

Difficulty: 3.

Show that every sequence tending to infinity has a minimum.

Difficulty: 3.

Show that every sequence tending to minus infinity has a maximum.

Difficulty: 3.

Suppose that \(\displaystyle a_n\to -\infty\), and let \(\displaystyle b_n=\max\{a_n,a_{n+1},a_{n+2},\ldots\}\). Show that \(\displaystyle b_n\to -\infty\).

Difficulty: 3.

Let \(\displaystyle a_n\) be a sequence and \(\displaystyle a\) be a number. What are the implications among the following statements?

a) \(\displaystyle \forall \varepsilon>0\) \(\displaystyle \exists N\) \(\displaystyle \forall n\geq N\) \(\displaystyle |a_n-a|<\varepsilon\).

b) \(\displaystyle \forall \varepsilon>0\) \(\displaystyle \exists N\) \(\displaystyle \forall n\geq N\) \(\displaystyle |a_n-a|\geq\varepsilon\).

c) \(\displaystyle \exists \varepsilon>0\) \(\displaystyle \forall N\) \(\displaystyle \forall n\geq N\) \(\displaystyle |a_n-a|<\varepsilon\).

d) \(\displaystyle \forall \varepsilon>0\) \(\displaystyle \forall N\) \(\displaystyle \forall n\geq N\) \(\displaystyle |a_n-a|<\varepsilon\).

e) \(\displaystyle \exists \varepsilon'>0\) \(\displaystyle \forall 0<\varepsilon<\varepsilon'\) \(\displaystyle \exists N\) \(\displaystyle \forall n\geq N\) \(\displaystyle |a_n-a|<\varepsilon\).

Difficulty: 3.

 *  [a)] \(\displaystyle a_n\to 1\). Does it imply that \(\displaystyle a_n^n\to 1\)?
 *  [b)] \(\displaystyle a_n>0, a_n\to 0\). Does it imply that \(\displaystyle \sqrt[n]{a_n}\to 0\)?
 *  [c)] \(\displaystyle a_n>0, a_n\to a>0\). Does it imply that \(\displaystyle \sqrt[n]{a_n}\to 1\)?
 *  [d)] \(\displaystyle c_nd_n\to 0\). Does it imply that \(\displaystyle c_n\to 0\) or \(\displaystyle d_n\to 0\)?

Difficulty: 3.

Which of the following statements is equivalent to the negation of \(\displaystyle a_n\to A\)? What is the meaning of the rest? What are the implications among them?
 (1)  For all \(\displaystyle \varepsilon >0\) there are infinitely many members of \(\displaystyle a_n\) outside of \(\displaystyle (A-\varepsilon,A+\varepsilon)\).
 (2)  There is an \(\displaystyle \varepsilon >0\), such that there are infinitely many members of \(\displaystyle a_n\) outside of \(\displaystyle (A-\varepsilon,A+\varepsilon)\).
 (3)  For all \(\displaystyle \varepsilon >0\) there are only finitely many members of \(\displaystyle a_n\) in the interval \(\displaystyle (A-\varepsilon,A+\varepsilon)\).
 (4)  There is an \(\displaystyle \varepsilon >0\), such that there are only finitely many members of \(\displaystyle a_n\) in the interval \(\displaystyle (A-\varepsilon,A+\varepsilon)\).

Difficulty: 3.

Is there a sequence of irrational numbers converging to (a) 1, (b) \(\displaystyle \sqrt 2 \)?

Difficulty: 3. Solution is available for this problem.

Give examples such that \(\displaystyle a_n-b_n\to 0\) but \(\displaystyle a_n/b_n\not\to1\), and \(\displaystyle a_n /b_n \to 1\) but \(\displaystyle a_n -b_n \not\to0\).

Difficulty: 3.

Does \(\displaystyle a_n^2\to a^2\) imply that \(\displaystyle a_n\to a\)? And does \(\displaystyle a_n^3 \to a^3\) imply that \(\displaystyle a_n \to a\)?

Difficulty: 3. Solution is available for this problem.

Prove that if \(\displaystyle a_n \to \infty\) and \(\displaystyle (b_n)\) is bounded, then \(\displaystyle (a_n +b_n )\to \infty.\)

Difficulty: 3.

Prove that if \(\displaystyle (a_n)\) has no subsequence tending to infinity, then \(\displaystyle (a_n)\) is bounded from above.

Difficulty: 3.

Prove that if \(\displaystyle a_n \to a >1,\) then \(\displaystyle (a_n^n ) \to \infty .\)

Difficulty: 3.

Prove that if \(\displaystyle (a_n +b_n )\) is convergent and \(\displaystyle (b_n )\) is divergent, then \(\displaystyle (a_n )\) is also divergent.

Difficulty: 3.

Is it true that if \(\displaystyle (a_n \cdot b_n )\) is convergent and \(\displaystyle (b_n )\) is divergent, then \(\displaystyle (a_n )\) is divergent?

Difficulty: 3.

Is it true that if \(\displaystyle (a_n /b_n )\) is convergent and \(\displaystyle (b_n )\) is divergent, then \(\displaystyle (a_n )\) is divergent?

Difficulty: 3.

Let \(\displaystyle \lim_{n\to \infty} a_n=a\), \(\displaystyle \lim_{n\to \infty} b_n=b\). Prove that \(\displaystyle \max(a_n,b_n)\to\max(a,b).\)

Difficulty: 3.

Prove that if \(\displaystyle y_n\to 0\) and \(\displaystyle Y=\lim {y_{n+1}\over y_n}\) exist, then \(\displaystyle y\in [-1,1]\).

Difficulty: 4.

Show that the sequence \(\displaystyle a_n\) is bounded if and only if for all sequences \(\displaystyle b_n\to 0\) the sequence \(\displaystyle a_nb_n\) also tends to \(\displaystyle 0\).

Difficulty: 4.

Give an example of a sequence \(\displaystyle a_n\to \infty\) such that \(\displaystyle \forall k=1,2,\ldots\) \(\displaystyle (a_{n+k}-a_n)\to 0\).

Difficulty: 4.

Give examples of sequences \(\displaystyle a_n\), with the property \(\displaystyle \displaystyle\frac{a_{n+1}}{a_n}\to1\), such that

 (1)  \(\displaystyle a_n\) is convergent;      (2)  \(\displaystyle a_n\to\infty\);  (3)  \(\displaystyle a_n\to-\infty\);      (4)  \(\displaystyle a_n\) is oscillating.

Difficulty: 4.

Show that every convergent sequence has a minimum or a maximum.

Difficulty: 4.

Give examples showing that if \(\displaystyle a_{n}\to 0\) and \(\displaystyle b_{n}\to +\infty\), then \(\displaystyle a_{n}b_{n}\) is critical.

Difficulty: 4.

Consider the sequence \(\displaystyle s_n\) of arithmetic means

\(\displaystyle s_n=\frac{a_1+\ldots +a_n}{n} \)

corresponding to the sequence \(\displaystyle a_n\). Show that if \(\displaystyle \lim\limits_{n\to \infty} a_n=a,\) then \(\displaystyle \lim\limits_{n\to\infty} s_n =a.\) Give an example when \(\displaystyle (s_n)\) is convergent, but \(\displaystyle (a_n)\) is divergent.

Difficulty: 4. Solution is available for this problem.

Consider the definition of \(\displaystyle a_n \to b\):

\(\displaystyle (\forall \varepsilon >0) (\exists n_0 ) (\forall n\ge n_0 )(|a_n -b |<\varepsilon ).\)

Changing the quantifiers and their order we can produce the following statements:
 (1)  \(\displaystyle (\forall \varepsilon >0) (\exists n_0 ) (\exists n\ge n_0 )(|a_n -b |<\varepsilon );\)
 (2)  \(\displaystyle (\forall \varepsilon >0) (\forall n_0 ) (\forall n\ge n_0 )(|a_n -b |<\varepsilon );\)
 (3)  \(\displaystyle (\exists \varepsilon >0) (\exists n_0 ) (\exists n\ge n_0 )(|a_n -b |<\varepsilon );\)
 (4)  \(\displaystyle (\exists n_0) (\forall \varepsilon >0 ) (\forall n\ge n_0 )(|a_n -b |<\varepsilon );\)
 (5)  \(\displaystyle (\forall n_0) (\exists \varepsilon >0 ) (\exists n\ge n_0 )(|a_n -b |<\varepsilon ).\) Which properties of the sequence \(\displaystyle (a_n )\) are expressed by these statements? Give examples of sequences (if they exist) satisfying these properties.

Difficulty: 4.

Consider the definition of \(\displaystyle a_n \to \infty\):

\(\displaystyle (\forall P) (\exists n_0 ) (\forall n\ge n_0 )(a_n >P ).\)

Changing the quantifiers and the orders we can produce the following statements:
 (1)  \(\displaystyle (\forall P) (\exists n_0 ) (\exists n\ge n_0 )(a_n >P );\)
 (2)  \(\displaystyle (\forall P) (\forall n_0 ) (\forall n\ge n_0 )(a_n >P );\)
 (3)  \(\displaystyle (\exists P) (\exists n_0 ) (\forall n\ge n_0 )(a_n >P );\)
 (4)  \(\displaystyle (\exists P) (\exists n_0 ) (\exists n\ge n_0 )(a_n >P );\)
 (5)  \(\displaystyle (\exists n_0) (\forall P ) (\forall n\ge n_0 )(a_n >P );\)
 (6)  \(\displaystyle (\forall n_0) (\exists P ) (\exists n\ge n_0 )(a_n >P ).\) Which properties of the sequence \(\displaystyle (a_n )\) are expressed by these statements? Give examples of sequences (if they exist) satisfying these properties.

Difficulty: 4.

Construct sequences \(\displaystyle (a_n)\) with all possible limit behavior (convergent, tending to infinity, tending to minus infinity, oscillating), while \(\displaystyle a_{n+1} -a_n \to 0\) holds.

Difficulty: 4.

Prove that if \(\displaystyle (a_{2n})\), \(\displaystyle (a_{2n+1})\), \(\displaystyle (a_{3n})\) are convergent, then \(\displaystyle a_n\) is convergent, too.

Difficulty: 4.

Prove that if \(\displaystyle a_n \to a,\) with \(\displaystyle |a|<1,\) then \(\displaystyle (a_n^n )\to 0.\)

Difficulty: 4.

Prove that if \(\displaystyle a_n \to a >0,\) then \(\displaystyle {\root n \of {a_n}} \to 1.\)

Difficulty: 4.

Let \(\displaystyle a_k\ne0\) and \(\displaystyle p(x)=a_0 +a_1 x+\ldots + a_k x^k\). Prove that

\(\displaystyle \lim\limits_{n\to +\infty}\frac{p(n+1)}{p(n)}=1. \)

Difficulty: 4. Solution is available for this problem.

Show that if \(\displaystyle a_n >0\) and \(\displaystyle a_{n+1} /a_n \to q,\) then \(\displaystyle \sqrt[n]{a_n}\to q.\)

Difficulty: 4.

Give an example of a positive sequence \(\displaystyle (a_n )\) for which \(\displaystyle \sqrt[n]{a_n}\to 1,\) but \(\displaystyle a_{n+1} /a_n \) does not tend to \(\displaystyle 1\). MegoEn E.g. merge \(\displaystyle b_n=1\) and \(\displaystyle c_n=n\).

Difficulty: 4.

Prove that if the sequence \(\displaystyle (a_n )\) has no convergent subsequence then \(\displaystyle |a_n |\to \infty .\)

Difficulty: 4. Solution is available for this problem.

Prove that if all subsequence of a sequence \(\displaystyle (a_n )\) have a subsequence tending to \(\displaystyle b\), then \(\displaystyle a_n \to b.\)

Difficulty: 4.

Does \(\displaystyle a_{n+1} -a_n \to 0\) imply that \(\displaystyle a_{2n}-a_n \to 0?\)

Difficulty: 4.

Give examples such that \(\displaystyle a_n \to \infty\) and

 (1)  \(\displaystyle a_{2n} -a_n \to 0;\)      (2)  \(\displaystyle a_{n^2} -a_n \to 0;\)      (3)  \(\displaystyle a_{2^n} -a_n \to 0;\)

Difficulty: 4.

Suppose that \(\displaystyle a_nb_n\to 1\), \(\displaystyle a_n+b_n\to 2\). Does it imply that \(\displaystyle a_n\to 1\), \(\displaystyle b_n\to 1\)?

Difficulty: 5.

Prove that if \(\displaystyle a_n\to\infty\), then \(\displaystyle \displaystyle\frac{a_1+a_2+\ldots+a_n}n\to\infty\).

Difficulty: 5.

Prove that if \(\displaystyle \forall n ~ a_n>0\) and \(\displaystyle a_n\to b\), then \(\displaystyle \root{n}\of{a_1a_2\ldots a_n}\to b\).

Difficulty: 5.

There are 8 possibilities for a sequence, according to monotonicity, boundedness and convergence. Which of these 8 classes are non-empty?

Difficulty: 5.

Assume that \(\displaystyle a_n\to a\) and \(\displaystyle a<a_n\) for all \(\displaystyle n\). Prove that \(\displaystyle a_n\) can be rearranged to a monotone decreasing sequence.

Difficulty: 5.

Prove that if the sequence \(\displaystyle (a_n )\) is bounded and all of its convergent subsequences tend to \(\displaystyle b\), then \(\displaystyle a_n \to b.\)

Difficulty: 5.

Prove that every sequence can be obtained as the product of a sequence tending to \(\displaystyle 0\), and a sequence tending to infinity.

Difficulty: 5.

Assume that \(\displaystyle a_n\to1\). What can we say about the limit of the sequence \(\displaystyle (a_n^n)\)?

Difficulty: 5.

How would you define \(\displaystyle 0^0\), \(\displaystyle \infty^0\) and \(\displaystyle 1^\infty\)? Explain it.

Difficulty: 5.

The sequence \(\displaystyle (a_n)\) satisfies the inequality \(\displaystyle a_n \le (a_{n-1} +a_{n+1} )/2\) for all \(\displaystyle n>1\). Prove that \(\displaystyle (a_n)\) cannot be oscillating.

Difficulty: 6.

Prove that if \(\displaystyle (a_n )\) is convergent and \(\displaystyle (a_{n+1} -a_n )\) is monotone, then \(\displaystyle n\cdot (a_{n+1} -a_n ) \to 0.\) Give an example for a convergent sequence \(\displaystyle (a_n )\) for which \(\displaystyle n\cdot (a_{n+1} -a_n ) \) does not tend to \(\displaystyle 0\).

Difficulty: 6.