Let f: N a t →R e ; f is a sequence (or array) which returns...

Let $f: N a t \rightarrow R e ; f$ is a sequence (or array) which returns values in Re. Define another function, $$ \max : \mathrm{Nat} \rightarrow \mathrm{Re} $$ as follows: For any $k \in N a t$, $$ \begin{gathered} \max (0)=f(0), \\ \text { and } \\ \max (k+1)=[\text { if } f(k+1)>\max (k) \text { then } f(k+1) \text { else } \max (k)], \end{gathered} $$ where $>$ is the greater than relation on the reals. Prove that: For any $j, n \in N a t$, if $j \leq n$, then $f(j) \leq \max (n)$. [Remark. One can implement this max function as an inefficient computer program. Since the value returned by $\max (n)$ is equal to the value returned by $f(j)$ for some $j \leq n$, this proof establishes that the program returns the maximum value of the array elements up to, and including, element $n$.]

Let $f: N a t \rightarrow R e ; f$ is a sequence (or array) which returns values in Re. Define another function,
$$
\max : \mathrm{Nat} \rightarrow \mathrm{Re}
$$
as follows: For any $k \in N a t$,
$$
\begin{gathered}
\max (0)=f(0), \\
\text { and } \\
\max (k+1)=[\text { if } f(k+1)>\max (k) \text { then } f(k+1) \text { else } \max (k)],
\end{gathered}
$$
where $>$ is the greater than relation on the reals. Prove that: For any $j, n \in N a t$, if $j \leq n$, then $f(j) \leq \max (n)$. [Remark. One can implement this max function as an inefficient computer program. Since the value returned by $\max (n)$ is equal to the value returned by $f(j)$ for some $j \leq n$, this proof establishes that the program returns the maximum value of the array elements up to, and including, element $n$.]

Key Concepts

Sequence

A sequence is an ordered list of elements, often defined as a function from the natural numbers to a set, such as the real numbers. In mathematics, sequences form the basis for discussing limits, convergence, and other properties, and they are widely used in various disciplines to model ordered data or events.

Recursive Function

A recursive function is one that is defined in terms of itself, typically with one or more base cases and a rule that relates the function's value at one point to its value at previous points. This concept is fundamental in both mathematics and computer programming as it allows for the definition of sequences, algorithms, and procedures that build upon simpler cases.

Mathematical Induction

Mathematical induction is a proof technique used to establish the truth of an infinite sequence of propositions. It involves proving a base case and then demonstrating that if a statement holds for an arbitrary case, it holds for the next case. This method is especially effective when dealing with recursively defined functions or properties of sequences.

Inequalities

Inequalities express relative sizes or orders between numbers, such as stating that one number is less than or equal to another. Understanding and manipulating inequalities is essential in many areas of mathematics, including analysis and optimization, and is crucial when proving that a function like the maximum function correctly captures the largest value in a sequence.

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