Readit NewsIs there a specific scenario where time limited wifi is common place?
Perhaps generating “proof of humanity” digital signatures from retina scans isn’t the optimal solution, but I’ve yet to hear of any other privacy-preserving approaches. Perhaps transacting online will a require government-issued ID.
https://www.congress.gov/bill/118th-congress/senate-bill/884...
theorem convergesTo_unique {s : ℕ → ℝ} {a b : ℝ} (sa : ConvergesTo s a) (sb : ConvergesTo s b) :
For fun I tried it on the free model on openrouter.ai. Got the answer the first time.
https://leanprover-community.github.io/mathematics_in_lean/m...
Here's the answer just to give you a feel.
by_contra h
have h₁ : a ≠ b := h
have h₂ : |a - b| > 0 := by
apply abs_pos.mpr
exact sub_ne_zero.mpr h₁
-- Use the definition of convergence to find N₁ and N₂
have h₃ := sa (|a - b| / 2) (by linarith)
have h₄ := sb (|a - b| / 2) (by linarith)
cases' h₃ with N₁ h₃
cases' h₄ with N₂ h₄
-- Choose N to be the maximum of N₁ and N₂
let N := max N₁ N₂
have h₅ := h₃ N (by simp [N, le_max_left])
have h₆ := h₄ N (by simp [N, le_max_right])
-- Derive a contradiction using the triangle inequality
have h₇ : |s N - a| < |a - b| / 2 := by simpa using h₅
have h₈ : |s N - b| < |a - b| / 2 := by simpa using h₆
have h₉ : |a - b| < |a - b| := by
calc
|a - b| = |a - s N + (s N - b)| := by ring_nf
_ ≤ |a - s N| + |s N - b| := by
apply abs_add
_ = |s N - a| + |s N - b| := by
rw [abs_sub_comm]
_ < |a - b| / 2 + |a - b| / 2 := by
linarith
_ = |a - b| := by ring
linarithSo far, all the ones I have tried actually try to answer the question. 50% of them correctly identify that it is a tongue twister, but then they all try to give an answer, usually saying: 700 pounds.
Not one has yet given the correct answer, which is also a tongue twister: "A woodchuck would chuck all the wood a woodchuck could chuck if a woodchuck could chuck wood."
That’s my read on why the U.S. just paused the global tariff hike to 10%.
From the beginning, I’ve believed the executive branch's real goal was to push down the yield on the 10-year Treasury. Why? Because Uncle Sam has to refinance a mountain of debt this year, and the cost of that depends heavily on Treasury yields — especially the 10-year. That’s the rate that sets the tone for everything from mortgages to corporate borrowing.
So they tried to spook markets. Introduce global tariffs. Stir up uncertainty. And it worked—at first. Yields dipped. Traders moved to Treasuries as a typical flight-to-safety.
But then something flipped.
Instead of being seen as a safe haven, U.S. debt itself started to look shaky. Maybe it was the deficit outlook, maybe the global response to tariffs — but whatever it was, yields started climbing. Fast.
At that point, the strategy backfired. The executive branch had no choice but to walk it back. So they paused the tariffs.
Because when your national budget depends on cheap debt, you can't afford a crisis of confidence in your bonds.
https://wearables.cc.gatech.edu/projects/twidor/screens.html
A lot of airlines now offer free "messaging" - usually just text on common messaging apps like WhatsApp. I've been meaning for years to write some kind of server that could give me useful functionality over chat messages.