I would like to prove the following,
Let $X$,$Y$ be infinite dimensional Banach-Spaces and $T$ a compact, linear and bounded operator. Then there exists a sequence $(x_n)_{n\in\mathbb N}$ with $||x_n||=1$ and $||Tx_n||\rightarrow\;0\quad n\rightarrow\infty$
Clearly if we consider a sequence with $||x_n||=1$ then $(||Tx_n||)_{n\in\mathbb N}$ contains a Cauchy subsequence and hence a converging subsequence.
Then by taking just the index set of the subsequence we have $||x_{n_k}||=1$ and $||Tx_{n_k}||\rightarrow\;y\quad k\rightarrow\infty$ for some $y\in TY$. But how to construct a sequence where the limit is 0? Could someone help me?
Thanks in advance!