### On a shadowing lemma in metric spaces

In the present paper conditions are studied, under which a pseudo-orbit of a continuous map $f:M\to M$, where $M$ is a metric space, is shadowed, in a more general sense, by an accurate orbit of the map $f$.

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In the present paper conditions are studied, under which a pseudo-orbit of a continuous map $f:M\to M$, where $M$ is a metric space, is shadowed, in a more general sense, by an accurate orbit of the map $f$.

In the paper, some kind of independence between upper metric dimension and natural order of converging sequences is shown — for any sequence converging to zero there is a greater sequence with an arbitrary ($\u2a7d1$) upper dimension. On the other hand there is a relationship to summability of series — the set of elements of any positive summable series must have metric dimension less than or equal to $1/2$.

Converging sequences in metric space have Hausdorff dimension zero, but their metric dimension (limit capacity, entropy dimension, box-counting dimension, Hausdorff dimension, Kolmogorov dimension, Minkowski dimension, Bouligand dimension, respectively) can be positive. Dimensions of such sequences are calculated using a different approach for each type. In this paper, a rather simple formula for (lower, upper) metric dimension of any sequence given by a differentiable convex function, is derived....

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