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 $$b_{2n-1}=a_n$$, $$b_{2n}=0$$.