1. On an island, there are 10 people, some of whom are honest people who always tell the truth, and others are liars who always lie. Each of them has thought of a real number (not necessarily an integer). Then, the first person says: “My number is greater than 1.” The second person says: “My number is greater than 2.” ... The tenth person says: “My number is greater than 10.” Afterwards, these ten people stand in a line in some order and say one by one: “My number is less than 1.” “My number is less than 2.” ... “My number is less than 10.” (Each person says exactly one sentence) How many honest people can there be at most among these people?
1. There can be at most 9 honest people.

Estimate: No honest person can say, "My number is greater than 10." (Otherwise, he would not be able to say any of the ten statements: "My number is less than 1," "My number is less than 2," ... "My number is less than 10.") Therefore, there must be at least 1 liar, and the number of honest people cannot exceed 9.
The following example shows that there can be 9 honest people.
Assume the first person thinks of the number 1.5, the second person thinks of the number 2.5, ... the ninth person thinks of the number 9.5, and the tenth person thinks of the number 5. In this way, the $k$-th person ($k = 1, 2, \cdots, 9$) can say, "My number is greater than $k$" and "My number is less than $k+1$" (indicating that they are all honest), while the tenth person says, "My number is greater than 10" and "My number is less than 1," meaning he is the liar.