This problem comes from Rudin's Functional Analysis, Chapter 10. It's the problem 15.
Suppose $X$ is a Banach space, $T\in\mathcal{B}(X)$ is compact, and $\|T^n\|\geq 1$ for all $n\geq 1$. Prove that the point spectrum of $T$ is not empty.
I know some useful facts but it's being hard to put them to work together. Here are some of them (always consider $\lambda\neq0$ and $\dim X = \infty$):
1) For all bounded sequence $(x_n)$ in $X$, there is a subsequence $(x_{n_k})$ such that $(Tx_{x_k})$ converges in $X$.
2) If $\lambda$ is in the spcetrum but not in the point spectrum, then $\ker(T-\lambda I) = \{0\}$ and Im$(T-\lambda I) \neq X$.
3) Im$(T-\lambda I)$ is closed in $X$.
4) $\sigma(T) = \sigma(T^\ast)$.
5) The spectrum is never empty.
6) Theorem 10.33 from the same book.
$H(\Omega)$ is the space of holomorphic function from $\Omega$ to $\mathbb{C}$ and $\tilde{f}(T) = \frac{1}{2\pi i}\int_\Gamma f(\zeta)(\zeta I - T)^{-1}\ d\zeta$ comes from symbolic calculus in Banach algebra.
Thank you for your help.