# Questions tagged [supremum-and-infimum]

For questions on suprema and infima. Use together with a subject area tag, such as (real-analysis) or (order-theory).

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### Infimum and Supremum of a set 2^k

I am trying to find the infimum and supremum of a set 2^k where k is an integer. I have determined that as k gets larger, so does 2^k so it is not bounded above and therefore there is no supremum. ...
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### Doubt in a step involving triangle inequality

This is a step in page 323 of the book Convex Optimization by Stephen Boyd. I am trying to understand how we move from 2nd to 3rd step. So far, I understood that we have to use triangle inequality and ...
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### How to prove the criteria for lower semicontinuity of a function at a point?

I have the following statement I need to prove. Let $f:D \to \mathbb{R}$. Then the function $f$ is lower semicontinuous at some point $x_0 \in D$ if $$\lim_{x \to x_0 } \inf f(x) \geq f(x_0)$$. I have ...
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### Greatest lower bound in Q [duplicate]

I have a set $$\{ r \in \mathbb Q \mid r^2 >2, r>0 \}$$ I was wondering why it does not have the greatest lower bound. Isn't $0 \in \mathbb Q$ a greatest lower bound for this set in rational ...
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### Proving that if a sequence $(a_n)$ is monotonic decreasing and $\lim a_n = l$, then $l=\inf \{a_n:n\in\mathbf{N}\}$

I am self-learning Real Analysis. In proving results about sequences and series of reals, it might useful to use the below fact, so I want to write a rigorous proof for it. But, I'd like to check if ...
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### Find infimum and closure of $A$

Let $$\mathbf{A}=\left\{\frac{\mathbf{3 m}+\mathbf{2 n}}{5\mathbf{m}+7 \mathbf{n}}: \mathbf{m}, \mathbf{n} \geq \mathbf{3}\right\}$$ Now find : $inf(A)$ $int(A)$ , interior points $cl(A)$ , ...
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### Let $K$ be a nonempty closed, convex subset of $R^d$. Prove that if $x\notin K$, then there exists a unique point $y\in K$ that is closest to $x$.
Let $K$ be a nonempty closed, convex subset of $R^d$. Prove that if $x\notin K$, then there exists a unique point $y\in K$ that is closest to $x$. My attempt: Suppose y and z are 2 different point ...