Question

Prove that a sequence that diverges to +∞ (resp. −∞) is divergent according to the previous definition

Answer 1

prev definition is not mentioned.

However trying to make sense from the question asked, it was figured, it is correctly formulated as a sequence converges to infinity. Then this sequence diverges i.e. doesn't converge.

So a sequence converges to infinity means given $M>0,$ there exists a natural number $N$

such that $\forall n\geq N, x_n>M.$ Here the sequence is denoted as $\{x_n\}.$ Now this sequence cannot converge to any point since if it converges to $x,$ then we take $\epsilon =1.$ So for some natural $N,$ $n\geq N\Rightarrow |x_n-x|<1.$ Hence we get, $x_n<x+1.$ This contradicts the previous definition by taking, $M=x+1.$ Hence the sequence diverges according to the definition that diverges means not convergent.