why use $A = \cap_{i=1}^{\infty} E_i ,E_{i+1} \subset E_{i}$ ,instead of $A=\lim_{i\to\infty}E_i$. In real analysis, why use $A = \cap_{i=1}^{\infty} E_i ,E_{i+1} \subset E_{i}$ ,instead of $A=\lim_{i\to\infty}E_i$.Are they the same things?
Is it because we prefer to use limits on real Numbers rather than on sets?

They are not the same. The condition you write implies the existence of the limit. But the converse does not hold. The limit can exists, e.g., also if you have an incresing sequence of sets.
