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Given a sequence of functions $f_n: (X,d_X) \rightarrow (Y,d_Y)$ where each function $f_n$ is bounded, I want show that if the $f_n$ converge uniform to some function $f:X \rightarrow Y$ then $f$ has to be bounded.

This seems quite obvious but I am not sure how to approach this. Does proof by contradiction work best ?

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    Yes indeed. Let $y \in Y$. Then there are $R_n$ such that $\forall n \in \mathbb N: f(X) \subseteq B(y,R_n)$. Let then $N \in \mathbb N$ such that for all x \in X: d_Y(f_N(x),f(x)) < 1. Then we get for all $x \in X$ that d_Y(f(x),y) \leq d_Y(f(x),f_N(x))+d_Y(f_N(x),y) < 1 + R_N and thus $f(X) \subseteq B(y,1+R_N)$.2012-12-03

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