As I understand, your question is about the equivalence for infinite dimensional Banach spaces of the two following statements:

a. $(x_{n}) \mbox{ in } X$ admits a convergent subsequence

b. $\displaystyle\liminf_{n} \|x_{n}\|<\infty$

If so, the answer is no. Certainly, in the finite dimensional case a. $\Leftrightarrow$ b.. In the infinite dimensional case, a. $\Rightarrow$ b.  but the converse is not true. Here a counter example: in $\ell_{2}$ take the sequence of the unitary vectors $(e_{n})$; we have $\liminf \|e_{n}\|=1<\infty$ but it does not admit any norm convergent subsequence because being $(e_{n})$ a weakly null sequence if a subsequence $(e_{n_{k}})$ were norm convergent it should be convergent to zero in the norm and this is not the case since $\|e_{n_{k}}\|=1$.

As I understand, your question is about the equivalence for infinite dimensional Banach spaces of the two following statements:

a. $(x_{n}) \mbox{ in } X$ admits a convergent subsequence

b. $\displaystyle\liminf_{n} \|x_{n}\|<\infty$

If so, the answer is no. Certainly, in the finite dimensional case a. $\Leftrightarrow$ b.. In the infinite dimensional case, a. $\Rightarrow$ b. but the converse is not true. Here a counter example: in $\ell_{2}$ take the sequence of the unitary vectors $(e_{n})$; we have $\liminf \|e_{n}\|=1<\infty$ but it does not admit any norm convergent subsequence because being $(e_{n})$ a weakly null sequence if a subsequence $(e_{n_{k}})$ were norm convergent it should be convergent to zero in the norm and this is not the case since $\|e_{n_{k}}\|=1$.

As I understand, your question is about the equivalence for infinite dimensional Banach spaces of the two following statements:

a. $(x_{n}) \mbox{ in } X$ admits a convergent subsequence

b. $\displaystyle\liminf_{n} \|x_{n}\|<\infty$

If so, the answer is no. Certainly, in the finite dimensional case a. $\Leftrightarrow$ b.. In the infinite dimensional case, a. $\Rightarrow$ b.  but the converse is not true. Here a counter example: in $\ell_{2}$ take the sequence of the unitary vectors $(e_{n})$; we have $\liminf \|e_{n}\|=1<\infty$ but it does not admit any norm convergent subsequence because being $(e_{n})$ a weakly null sequence if a subsequence $(e_{n_{k}})$ were norm convergent it should be convergent to zero in the norm and this is not the case since $\|e_{n_{k}}\|=1$.

As I understand, your question is about the equivalence for infinite dimensional Banach spaces of the two following statements:

a. $(x_{n}) \mbox{ in } X$ admits a convergent subsequence

b. $\displaystyle\liminf_{n} \|x_{n}\|<\infty$

If so, the answer is no. Certainly, in the finite dimensional case a. $\Leftrightarrow$ b.. In the infinite dimensional case, a. $\Rightarrow$ b.  but the converse is not true. Here a counter example: in $\ell_{2}$ take the sequence of the unitary vectors $(e_{n})$; we have $\liminf \|e_{n}\|=1<\infty$ but it does not admit any norm convergent subsequence because being $(e_{n})$ a weakly null sequence if a subsequence $(e_{n_{k}})$ were norm convergent it should be convergent to zero in the norm and this is not the case since $\|e_{n_{k}}\|=1$.