Let {an} be a sequence with an --> a, and define a new sequence
{bn} by bn = max{an, an+1, an^2}, prove that bn --> a

* an+1 is increment of index n, an^2 the root is only on the index n as well.

any help will be much appreciate.

I think I got it, bn = max{an... an^2} = ak, k>=n
and from the limit definition, n >= N and |an-a| < ε, so if k >= n >= N, then also |bn-a| < ε, then bn --> a, as well... am I correct?