Consider a compact Riemannian manifold endowed with a Morse function f. Fix two critical points x and y of f. Whenever you have a sequence $u_k$ of Morse trajectories connecting x and y it might happen that the $u_k$ "converge" to a broken trajectory. Suppose there is a bound $\|u_k'(s)\|\le C$ independently of k and s. Can the sequence still break?
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