Sequences: Convergence, Monotone Convergence Theorem, and the Squeeze Theorem

A sequence is the discrete analogue of a function: rather than $f(x)$ for real $x$, it gives a number $a_n$ for each positive integer $n$. Whether a sequence converges — whether $a_n$ settles to a finite limit $L$ — is the first question in the theory of infinite series and in any iterative algorithm. In finance, sequences model the evolution of discrete time-steps: the balance of a savings account after $n$ compounding periods, the price of a bond with $n$ coupon dates remaining, the $n$-period binomial tree price. The Monotone Convergence Theorem (bounded and monotone implies convergent) underpins the existence of the Black-Scholes price as a limit of binomial tree prices as $n \to \infty$.

Learning objectives. By the end of this note you should be able to: