Let such that . Prove that .

# real analysis

# One step forward, two steps back

Let be an infinitely differentiable function. Does there existe also infinitely differentiable such that for all ?

# One moment please

Let be continuous functions. We call the *-th moment* of . Prove that if and have identical -th moments for all , then .

# Constant Function

Let be a continous and bounded function such that for all

Then is constant.

# Wronskians and linear dependence

If are sufficiently differentiable, their *Wronskian* is defined as

Obviously, if are linearly dependent, then their Wronskian vanishes identically on , and the converse is true if the functions are analytic. Find non-analytic functions for which the converse fails.

# Balanced signs

Assume is a nonincreasing sequence of positive integers whose sum diverges and assume signs are chosen so that the modified series converges. Then

Assume now only the signed series converges, with no hypothesis on the unsigned sequence. Show

# I know one when I see one

Suppose is such that for each , is eventually zero. Then is a polynomial.