Question

Use the definition of limit to prove that the sequence {n − 1 }∞n=1 is divergent.

Answer 1

$Solution: Using ~the ~definition~ we~have, Let~a_n={n-1} \\ \lim_{n \to \infty} a_n = \lim_{n \to \infty} {n-1}=L \\there~ exists~ an ~\epsilon~such~that~for~all~N>0,there~is~an~n>N~with~ \\|a_n-L| \geq ~\epsilon~~~~~................................................(1) \\Since~ when~ we~ put~n=1,2,3,4,....... \\we~get~sequence~\{-1,0,1,2,3,4,......\} \\ when ~we~put~\lim_{n \to \infty} ~it ~approches~ to ~\infty. \\\therefore , by ~(1),~sequencce~\{a_n\} ~diverges.$