Problem 19
Theorem 4 shows that a convergent series does not change its sum if every several consecutive terms are replaced by their sum. Show by an example that the reverse process (splitting each term into several terms) may affect convergence.
Problem 19
Prove that if \(X\) and \(Y\) are connected, so is \(X \times Y\) under the product metric.
Problem 20
Find \(\sum_{n=1}^{\infty} \frac{1}{n(n+1)}\)
Problem 21
The functions \(f_{n}: A \rightarrow\left(T, \rho^{\prime}\right), A \subseteq(S, \rho)\) are said to be equicontinuous at \(p \in A\) iff $$ (\forall \varepsilon>0)(\exists \delta>0)(\forall n)\left(\forall x \in A \cap G_{p}(\delta)\right) \quad \rho^{\prime}\left(f_{n}(x), f_{n}(p)\right)<\varepsilon $$ Prove that if so, and if \(f_{n} \rightarrow f\) (pointwise) on \(A\), then \(f\) is continuous at \(p\).
