A limit describes the entire eventual tail of a sequence.
WHAT HAPPENS IF…Does staying between −1 and 1 imply a limit exists?
Interactive exploration
Loading the mathematical model…
Connect the picture to the ideaan→L⟺∀ε>0∃N∀n>N:∣an−L∣<ε
For aₙ=1+1/n, the error is 1/n, so n>1/ε is sufficient. The alternating sequence (−1)ⁿ never stays near zero: its even and odd subsequences remain at 1 and −1. A bounded sequence therefore need not converge.
A detail that matters
A few late terms near a candidate are not enough; the entire tail must eventually stay near it.