You can define a method/algorithm stable if during the various steps it not amplifies excessively the errors on the data.
So you introduce some bounded estimate.
Note that this kind of errors, monitored by the stability check, are different from an error occurred by the ill-condition of a problem,
i.e. in the stability you check the errors propagated by the algorithm itself.
Another thing to note is that stability is connected to the problem, for example the same method is stable for a problem and unstable for a different problem,
see chapter 1.16 of .
As reference can be . The book of Kendal, , cover different arguments and you can find the relative stability proof (I suggest it because I like it). In general books over specific argument have got the stability proof for the methods covered.
 Nicholas J. Higham, MR 1927606 Accuracy and stability of numerical algorithms, ISBN: 0-89871-521-0.
 Atkinson, Kendall, and Weimin Han. Theoretical numerical analysis. Vol. 39. Berlin: Springer, 2005.