Convergent Real Sequence/Examples/x (n+1) = k over 1 + x n/Mistake

Source Work

1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach:

Chapter $\S 5$: Subsequences

$\S 5.7$: Solution to Exercise $(3)$

This mistake can be seen in the $1977$ edition as published by Cambridge University Press: ISBN 0 521 29167 4

Mistake

It is easily shown by induction that $k > x_n > 0$ ($n = 1, 2, \ldots$).

Correction

The condition does not necessarily hold for $n = 1$.

$x_1$ is specified as being (strictly) positive, but it is not stated (and indeed, it is not necessary) that $x_1 < k$.

Hence the line should read:

It is easily shown by induction that $k > x_n > 0$ ($n = 2, 3, \ldots$).

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 5$: Subsequences: Exercise $\S 5.7 \ (3)$