Minimizing sequence

A sequence of elements $ y _ {n} $ from a set $ M $ for which the corresponding sequence of function values $ \phi ( y _ {n} ) $ tends to the greatest lower bound of $ \phi $ on $ M $, that is,

$$ \lim\limits _ {n \rightarrow \infty } \phi ( y _ {n} ) = \ \inf _ {y \in M } \phi ( y) . $$

The compactness of a minimizing sequence, that is, the existence of a subsequence converging to an element of $ M $, in combination with the lower semi-continuity of $ \phi $, guarantees the existence of an optimal element

$$ y ^ {n} \in M ,\ \phi ( y ^ {n} ) = \min _ { y \in M } \phi ( y) . $$

In approximation theory, a minimizing sequence $ \{ y _ {n} \} \in M $ for a given element $ x $ of a metric space $ X = ( X , \rho ) $ is a sequence for which

$$ \rho ( x , y _ {n} ) \rightarrow \rho ( x , M ) = \inf \ \{ {\rho ( x , y ) } : {y \in M } \} . $$

See Approximately-compact set.