The ε-N definition of sequence limit is really hard to understand. Only by doing more questions can we gradually understand the connotation of the definition.

N=[ 1/ε]+ 1 is taken to ensure that the absolute value of the difference between the nth term and all subsequent terms and 2 (actually, 1/n) is less than ε.

So take N=[ 1/ε]+2, and N=[ 1/ε]+3. . . Both will do.

As for adding one more 1 instead of directly taking N=[ 1/ε], it is mainly to satisfy | an-2 |.

Actually, because [1/ε] < = 1/ε, actually1/[1/ε] > = 1/( 1/ε)=ε ,

After adding 1, n = [1/ε]+1> 1/ε, so 1/n