In this paper we consider Bernstein's Lethargy Theorem (BLT) in the context
of Fr\'{e}chet spaces. Let X be an infinite-dimensional Fr\'echet space and
let V={Vn} be a nested sequence of subspaces of X such that Vnˉ⊆Vn+1 for any n∈N and X=⋃n=1∞Vnˉ. Let en be a decreasing sequence of
positive numbers tending to 0. Under an additional natural condition on
\sup\{\{dist}(x, V_n)\}, we prove that there exists x∈X and no∈N such that \frac{e_n}{3} \leq \{dist}(x,V_n) \leq 3 e_n for
any n≥no. By using the above theorem, we prove both Shapiro's
\cite{Sha} and Tyuremskikh's \cite{Tyu} theorems for Fr\'{e}chet spaces.
Considering rapidly decreasing sequences, other versions of the BLT theorem in
Fr\'{e}chet spaces will be discussed. We also give a theorem improving
Konyagin's \cite{Kon} result for Banach spaces.Comment: 20 page