31-34 True-False Determine whether the statement is true or false. Explain your answer. 31. Sequences are functions. 32. If \( \left\{a_{n}\right\} \) and \( \left\{b_{n}\right\} \) are sequences such that \( \left\{a_{n}+b_{n}\right\} \) converges, then \( \left\{a_{n}\right\} \) and \( \left\{b_{n}\right\} \) converge. 33. If \( \left\{a_{n}\right\} \) diverges, then \( a_{n} \rightarrow+\infty \) or \( a_{n} \rightarrow-\infty \). 34. If the graph of \( y=f(x) \) has a horizontal asymptote as \( x \rightarrow+\infty \), then the sequence \( \{f(n)\} \) converges.

31. True. Sequences are indeed a specific type of function that assigns each natural number (or integer) to a real number. Each term of a sequence can be thought of as the output of a function defined on the set of natural numbers. 32. False. The convergence of the sequence \( \{a_n + b_n\} \) does not imply that both \( \{a_n\} \) and \( \{b_n\} \) must converge. It's possible for one of them to diverge in a manner that balances the other, leading to convergence in their sum. 33. False. While a diverging sequence does not converge to a finite limit, it can also exhibit behavior that does not trend toward \( +\infty \) or \( -\infty \). For example, it could oscillate indefinitely without approaching any specific values. 34. True. If the graph of \( y = f(x) \) has a horizontal asymptote as \( x \rightarrow +\infty \), it signifies that \( f(x) \) approaches a particular constant value. Consequently, the same holds for the sequence \( \{f(n)\} \) as \( n \rightarrow +\infty \), meaning it will converge to that constant value.