SKU: 65395696935

The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs

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Description

The Real Analysis Lifesaver: All the Tools You Need to Understand ProofsThe essential lifesaver that every student of real analysis needs Real analysis is difficult. For most students, in addition to learning new material about real numbers, topology, and sequences, they are also learning to read and write rigorous proofs for the first time. The Real Analysis Lifesaver is an innovative guide that helps students through their first real analysis course while giving them the solid foundation they need for further study in

The essential lifesaver that every student of real analysis needs

Real analysis is difficult. For most students, in addition to learning new material about real numbers, topology, and sequences, they are also learning to read and write rigorous proofs for the first time. The Real Analysis Lifesaver is an innovative guide that helps students through their first real analysis course while giving them the solid foundation they need for further study in proof-based math.

Rather than presenting polished proofs with no explanation of how they were devised, The Real Analysis Lifesaver takes a two-step approach, first showing students how to work backwards to solve the crux of the problem, then showing them how to write it up formally. It takes the time to provide plenty of examples as well as guided fill in the blanks exercises to solidify understanding.

Newcomers to real analysis can feel like they are drowning in new symbols, concepts, and an entirely new way of thinking about math. Inspired by the popular Calculus Lifesaver, this book is refreshingly straightforward and full of clear explanations, pictures, and humor. It is the lifesaver that every drowning student needs.

  • The essential "lifesaver" companion for any course in real analysis
  • Clear, humorous, and easy-to-read style
  • Teaches students not just what the proofs are, but how to do them--in more than 40 worked-out examples
  • Every new definition is accompanied by examples and important clarifications
  • Features more than 20 "fill in the blanks" exercises to help internalize proof techniques
  • Tried and tested in the classroom


Binding Type: Paperback
Publisher: Princeton University Press
Published: 01/10/2017
ISBN: 9780691172934
Pages: 200
Weight: 1.10lbs
Size: 9.90h x 6.90w x 0.60d

Review Citations: Choice 07/01/2017
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SKU: 65395696935

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M
Birmingham, US
★★★★★ 5
Good for night sessions
Size: Medium, Pattern Name: Pack of 1
Picked this up for evening fetch sessions and it’s been a win. It charges up fast, couple minutes under a lamp or in sunlight and it’s good to go, and the glow is genuinely bright, not the sad dim purple some of these put out. Makes it way easier to find in the yard at dusk, and the dog can actually track it. It’s a little on the softer side compared to a regular tennis ball, but durable enough to hold up to my big dog, who is not gentle with toys. Still kicking after plenty of chomping and slobber. Fits a standard Chuckit! launcher perfectly, which is the whole point. Pros: • Glows brightly and clearly in the dark • Charges fast from any light source • Holds up well even with a big, enthusiastic chewer • Fits standard ball-throwing sticks • Great for evening or early-morning fetch Cons: • A bit softer than a regular tennis ball — a determined power-chewer could probably destroy it eventually • Glow fades after a while, so you’ll need to recharge it during long sessions • Only glows if it’s been charged, which is obvious but easy to forget Tips: • Charge it under a bright light or in the sun for a few minutes before heading out • A flashlight works in a pinch if you forgot to charge it • Keep it out of permanent fetch duty — rotate with a regular ball so it lasts longer • Not a chew toy, even though it’s tempting to leave it with the dog Five stars. If you’ve got a dog and a yard, this makes evening fetch actually doable.
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Reviewed in the United States on May 24, 2026
V
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VOKLST
Louisville, US
★★★★★ 5
Westies Fav
Size: Small, Pattern Name: Pack of 1
Best Westie ball ever, my dog loves this ball. Exactly her size! 3" tennis balls are too big & she rips the fur off. this silicone-ish ball has no fur & is 2&1/2 " We have 2" balls from her electronic ball laucher, but she still loves this glow in the dark best!
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Reviewed in the United States on May 4, 2026
B
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Book Junkie
Lowell, US
★★★★★ 5
Best dog ball in the world!
Size: Large, Pattern Name: Pack of 1, Size: Large, Pattern Name: Pack of 1
My dog is singularly obsessed with playing ball. He would drop a steak to chase a ball. (He'd return to the steak, but not without the retrieved ball. He's not an idiot.) He makes dangerous leaps, crashes into trees, and runs fast enough to catch thrown balls before they even hit the ground. Since he doesn't care about his own wellbeing, it's my job to protect him as much as I can. And a lot of balls are hard enough to cause him pain or dental damage when he mistimes a catch. Enter the ChuckIt Glow Ball! The minute Cowboy got his first Glow Ball, he no longer had any interest in the dozens of tennis balls scattered around the house and yard. If I took the Glow Ball away and instead threw a tennis ball, Cowboy would run to retrieve the ball, then return without it and begin looking all over the house for the Glow Ball. The Glow Ball is tough enough to survive Cowboy's big, sharp chompers, yet it's soft enough to do no harm when it hits him in the face. The glow feature is a huge bonus, too. Cowboy gets distracted by chipmunks and squirrels when we're playing, so he sometimes can't remember where he dropped the ball to pursue the interloper. If the ball was exposed to enough light before it was misplaced, it will be easy to find once the sun goes down. (Otherwise, Cowboy always finds it eventually.) When I'm feeling nocturnal, I'll put the ball under a light for a few minutes to activate the electrons, then treat my boy to a middle-of-the-night game of fetch. I was so happy to see these on sale, I bought a lifetime supply for my pantry. The size guide seems accurate. ChuckIt recommended the large size for a dog of Cowboy's weight and that size is perfect. Highly recommended product.
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Reviewed in the United States on October 14, 2024
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jewelrycook
Whiting, US
★★★★★ 5
Bouncy, not hard, and good value.
Style: Fetch Balls (Pack of 10), Size: 2.5 inch
Great balls for the ball launcher and not rockhard like some of the solid rubber balls. These have great bounce, and because of the two holes, they make a fun whistling noise as they fly through the air. My dogs love them and they are a good value. By the way, if you throw them too hard, they will go through chain-link and no climb fence because they squish a bit!
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Reviewed in the United States on February 16, 2026
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IT Secure Force
San Leandro, US
★★★★★ 5
Excellent dog balls — safe design and very durable
Style: Fetch Balls (Pack of 3), Size: 2.5 inch
These fetch balls from Amazon Basics are fantastic, and my dog absolutely loves them. One of the biggest reasons I prefer these over other brands is the hole that runs through the center of the ball. This is a very important safety feature: if a dog were to accidentally get the ball stuck in the back of the throat, the hole allows air to pass through. That small detail could literally make a life-saving difference. Aside from the safety benefit, they bounce well, are easy to clean, and hold up to a lot of playtime. They’re firm but not rock-hard, and my dog goes crazy for them. Great value, great quality, and a thoughtful design. Highly recommended for dog owners who want a safer alternative to standard solid rubber balls.
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Reviewed in the United States on December 1, 2025

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