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.