Statements about limits
Task number: 2864
Decide whether the following statements are valid. If so, prove them. If not, first disprove them and then try to modify them to be true (if possible).
Variant 1
\(\displaystyle \lim_{n\to \infty} a_n= a \Longleftrightarrow \lim_{n\to \infty} a_{n+1}=a \)
Variant 2
\(\displaystyle \lim_{n\to \infty} a_n= a \Longleftrightarrow \lim_{n\to \infty} a_{2n}=a \)
Variant 3
\(\displaystyle (\exists n_0\in \mathbb N\ \forall n\ge n_0: a_n \le b_n) \Longrightarrow \lim_{n\to \infty} a_n \le \lim_{n\to \infty} b_n \)
Variant 4
\(\displaystyle (\exists n_0\in \mathbb N\ \forall n\ge n_0: a_n < b_n) \Longrightarrow \lim_{n\to \infty} a_n < \lim_{n\to \infty} b_n \)
Variant 5
\(\displaystyle \lim_{n\to \infty} a_n < \lim_{n\to \infty} b_n \Longrightarrow (\exists n_0\in \mathbb N\ \forall n\ge n_0: a_n < b_n) \)
Variant 6
\(\displaystyle \lim_{n\to \infty} a_n =a \Longleftrightarrow \lim_{n\to \infty} b_n =a \)
for the sequence \(b_n\) defined by \(b_{2n-1}=a_n\), \(b_{2n}=0\).
