For which values of \(\displaystyle x\) do the following sequences converge? On which intervals do they converge uniformly?

\(\displaystyle \root{n}\of{|x|} \qquad \frac{x^n}{n!} \qquad x^n-x^{n+1} \qquad \left(1+\frac{x}n\right)^n \)

Difficulty: 3.

A sequence of functions \(\displaystyle f_1,f_2,\ldots:I\to\R\) is uniformly bounded, if \(\displaystyle \exists K\in\R ~ \forall n\in\N ~ \forall x\in I ~ |f_n(x)|<K\).

Prove that the limit of a uniformly bounded sequence of functions is bounded.

Difficulty: 3.

For which values of \(\displaystyle x\) do the following sequences converge? On which intervals do they converge uniformly?

\(\displaystyle \frac{x^n}{1+x^n} \quad \root{n}\of{1+x^{2n}} \quad \sqrt{x^2+\frac1n} \)

Difficulty: 3.

True or false?

(a) A pointwise limit of monotonic functions is monotonic?

(b) A pointwise limit of strictly monotonic functions is strictly monotonic?

(c) A pointwise limit of bounded functions is bounded?

(d) A pointwise limit of continuous functions is continuous?

(e) A pointwise limit of Lipschitz functions is Lipschitz?

Difficulty: 4.

True or false?

(a) A uniform limit of monotonic functions is monotonic?

(b) A uniform limit of strictly monotonic functions is strictly monotonic?

(c) A uniform limit of bounded functions is bounded?

(d) A uniform limit of continuous functions is continuous?

(e) A uniform limit of Lipschitz functions is Lipschitz?

Difficulty: 4.

True or false?

(a) A pointwise limit of convex functions is convex.

(b) A pointwise limit of strictly convex functions is strictly convex.

(c) A pointwise limit of Riemann-integrable functions is Riemann-integrable.

(d) A pointwise limit of differentiable functions is differentiable.

Difficulty: 4.

True or false?

(a) A uniform limit of convex functions is convex.

(b) A uniform limit of strictly convex functions is strictly convex.

(c) A uniform limit of Riemann-integrable functions is Riemann-integrable.

(d) A uniform limit of differentiable functions is differentiable.

Difficulty: 4.

True or false? If a sequence of continuous functions \(\displaystyle f_n:[a,b]\to\R\) uniformly convergent on \(\displaystyle [a,b]\cap\Q\) then it is uniformly convergent on \(\displaystyle [a,b]\).

Difficulty: 5.

A sequence of functions \(\displaystyle f_1,f_2,\ldots:I\to\R\) is uniformly Lipschitz, if \(\displaystyle \exists K\in\R ~ \forall n\in\N ~ \forall x,y\in I ~ |f_n(x)-f_n(y)|\le K|x-y|\). Prove that a pointwise limit of a sequence of uniformly Lipschitz functions is Lipschitz.

Difficulty: 5.

True or false? If \(\displaystyle f_1,f_2,\ldots\) is a sequence of continuous non-negative functions, then \(\displaystyle F(x)=\inf\{f_1(x),f_2(x),\dots\}\) is also continuous.

Difficulty: 5.

Prove that \(\displaystyle \zeta(s)\) is infinitely differentiable on \(\displaystyle (1,\infty)\).

Difficulty: 6.

Prove that a uniformly bounded and uniformly Lipschitz sequence of functions has a uniformly convergent subsequence.

Difficulty: 7.

Prove that if \(\displaystyle (f_n:H \to \R)\) is uniformly convergent on all countable subsets of \(\displaystyle H\), then it is uniformly convergent on \(\displaystyle H\).

Difficulty: 7.

True or false? From a sequence of uniformly bounded continuous functions \(\displaystyle f_n:[a,b]\to \R\) one can select a uniformly convergent subsequence.

Difficulty: 9.

True or false? If \(\displaystyle H\) is a nonempty bounded and closed subset of \(\displaystyle C[a,b]\) and \(\displaystyle f:H\to\R\) is a continuous map, then \(\displaystyle f\) has a maximum.

Difficulty: 9.

Is the Baire theorem true for \(\displaystyle C[a,b]\)? That is, decide whether \(\displaystyle C[a,b]\) can be presented as a union of countably many nowhere dense subsets.

Difficulty: 9.