The Direct Method of the Calculus of Variations

A real function φ of a real variable which is bounded below on the real line needs not to have a minimum, as it is clear from the example of the exponential function. If we call minimizing sequence for φ any sequence (a k ) such that $$ \varphi ({a_k}) \to \inf \varphi $$ as k → ∞, a necessary condition for the real number a to be such that $$ \varphi (a) \to \inf \varphi $$ is that φ has a minimizing sequence which converges to a (take a k = a for all integers k). Without suitable continuity assumptions on φ this condition will not be sufficient, as shown by the example of the function φ defined by φ(x) = |x| for x ≠ 0 and φ(0) = 1, which does not achieve its infimum 0 although all its minimizing sequences converge to zero. In order that the limit a of a convergent minimizing sequence be such that φ(a) = inf φ, we have to impose that $$ \mathop {\lim }\limits_{k \to \infty } \varphi ({a_k}) \ge \varphi (a) $$