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Section Summary
Important ideas that we discussed in this section include the following.
A sequence in a metric space \(X\) is a function \(f : \Z^+ \to X\text{.}\)
A sequence \((a_n)\) in a metric space \((X,d)\) has a limit \(L\) in \(X\) if, given any \(\epsilon \gt 0\) there exists a positive integer \(N\) such that \(d(a_n,L) \lt \epsilon\) whenever \(n \geq N\text{.}\)
Let \(f\) be a function from a metric space \((X,d_X)\) to m metric space \((Y,d_Y)\text{.}\) Then \(f\) is continuous at \(a \in X\) if and only if \(\lim f(a_n) = f(a)\) for any sequence \((a_n)\) in \(X\) that converges to \(a\text{.}\)
