n/(n−1) = 1 + 1/(n−1) → 1 as n → ∞.
Hence aₙ → 1³ = 1, a finite unique limit.
The sequence is monotonically decreasing (a₂ = 8, a₃ = 3.375, a₄ = 2.37 …) and tends to 1, so it converges.
It is also bounded, but “converging” is the stronger and intended description; a convergent sequence is necessarily bounded, whereas bounded alone would not exclude oscillation.
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