Very short proof that limit preserving functions (continuous functions) on complete partial orders are necessarily monotone.

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## Posts Tagged ‘ analysis ’

# Even Pascals

Let , where k is an integer. Prove that for any positive integer n the number

is divisible by .

# Sequences in Sequences

Define the sequence recursively by and

Find an explicit formula for in terms of n.

# More Linear Recurrences

Let be defined by the recurrence relation , with . Show that the expression depends only on b and , but not on a.

# Almost Linear

Find the general term of the sequence given by , and

# Polynomial Divisors

Let . Show that for any positive integers there exists unique numbers such that the polynomial is divisible by p(x)

# Analytical Fibonacci

We derive Binet’s equation for the n^{th} Fibonacci number as

# Linear Recursive Sequence

We derive a general technique for solving full rank linear recursive sequences. Formally, a k^{th} linear recursive sequence is defined as

# The Fib-Fib-Fib Sequence

Consider the sequences , defined recursively by

Show that

# Square Sequence

The sequence satisfies

for all nonnegative integers n,m and . If , determine