Continuity from complete metric space Let $f:X\rightarrow Y$ be a continuous function, such as: $f(X)=Y$.
If $X$ is complete, does it imply $Y$ is complete?

No. Let $X = Y = \mathbb{N}$ with metrics
$$d_X(m, n) = |m - n|$$
and
$$d_Y(m, n) = |2^{-m} - 2^{-n}|.$$
The identity map is a continuous bijection from $X$ to $Y$, $X$ is complete, but $Y$ is not (because $0, 1, 2, 3, \ldots$ is a Cacuhy sequence without a limit point).
