How to deduce the result $\zeta(2k)_{k\to \infty} \to 1$

Question

While studying Analytic number theory from Tom M Apostol Introduction to analytic number theory , theorem 12.18 which is related to Bernoulli Numbers I cannot deduce how Apostol writes -

$\zeta(2k)_{k\to \infty} \to 1$ Can it be deduced from this result

If yes, can someone please tell how

Answer 1

As @RaymondManzoni notes, that's not Apostol's strategy. It's easiest to prove $\lim_{s\to\infty}\zeta(s)=1$ by the squeeze theorem viz.$$n^{-s}\le\int_{n-1}^nx^{-s}ds\implies\zeta(s)\le1+\int_1^\infty x^{-s}ds=1+\frac{1}{s-1}$$for $s>1$.