# Tagged Questions

Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, integration through the fundamental theorem of calculus.

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### Compact and non-compact curves in the plane

I want to find compact and non-compact curves in the plane: MY attempt" Suppose $f: \mathbb{R} \to \mathbb{R}$ is continous function. Put $C = \{ (x, f(x) ) \}$. Then $C$ is closed. IS it bounded? ...
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### An exotic sequence

Let $a=\frac{1+i\sqrt 7}{2}$ and $u_n=\Re(a^n)$ show that $(|u_n|)\to +\infty$ I think basics method does not works here. Any ideas ?
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### help with showing completeness

Let $\left\{H_n\right\}_{n=1}^\infty$ be a sequence of Hilbert spaces and let $H=\left\{\left\{x_n\right\}:x_n\in H_n, \sum ||x_n||^2<\infty \right\}$. Define the inner product as ...
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### Proving a certain map on the closed unit disc must be the identity

Re-bountied! Prove: Let $f$ be a continuous function on the closed unit disc with two properties: 1. $f$ is the identity on the boundary, i.e., on the unit circle. That is, if $|z| = 1$, then ...
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### Proof of L'Hôpital's Rule for $x \to$ finite $a^{+}$ (J Stewart pp A-46)

Reduce all cases of indeterminate form to $x \to 0^{+}$ and $f(x),g(x) \to 0$. But Stewart proves it for $x \to finite \; a^{+}$. I don't know why? Ergo in place of $x \to 0^{+}$, I'll chagrin about ...
Let a partition $\{t_0,\ldots,t_n\}$ of the interval $[a,b]$ and let $f$ an integrable function. (we may also assume that $f$ is differentiable on $[a,b]$) We know that the Right Riemann sum is ...
I have just started studying fourier series. All the convergences I have seen considered the partial sums to be $\sum\limits_{i=-n}^n a_n Sin(n\theta)$. But in all practical systems the harmonics ...