If we define a continuous real function $f: M \to R$ as one that preserves sequential convergence: $$ (x_n)\to p \Rightarrow f(x_n) \to f(p), \tag{1} $$ then to show that this definition is equivalent to the epsilon-delta condition, $$ \forall p \forall \epsilon \exists \delta: x \in V_{\delta}(p) \Rightarrow f(x) \in V_{\epsilon}(f(p)), \tag{2} $$ my textbooks use use a proof by contradiction in which they derive a convergent sequence that does not converge under the mapping.

Question: Is it possible to have a direct proof of $(1) \to (2)$?

Thought: I think to directly show that $(1) \to (2)$, we will have to show that every element of M is an element of at least a sequence in $M$, i.e., the union of the elements of all the sequences in $M$ is $M$.


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